Prove: (sec theta - 1) (sec theta + 1) = tan ^2 theta

(sec theta - 1) (sec theta + 1); = sec ^2 theta - 1 Algebra; = tan ^2 theta Pythagorean identity

Prove: (1 + tan x)/(sec x) = cos x + sin x

(1 + tan x)/(sec x); = cos x (1 + tan x) Reciprocal identity; = cos x + cos x tan x Algebra; = cos x + cos x × (sin x)/(cos x) Quotient identity; = cos x + sin x Algebra

Prove: csc x - sin x = cos x cot x

csc x - sin x; = (1)/(sin x) - sin x Reciprocal identity; = (1 - sin ^2 x)/(sin x) Algebra; = (cos ^2 x)/(sin x) Pythagorean identity; = cos x × (cos x)/(sin x) Algebra; = cos x cot x Quotient identity

Prove: (sin theta + cos theta)^2 = 1 + 2sin theta cos theta

(sin theta + cos theta)^2; = sin ^2 theta + 2sin theta cos theta + cos ^2 theta Algebra; = 1 + 2sin theta cos theta Pythagorean identity

Give the reason for each numbered step.; [3pt] tan beta + cot beta; = (sin beta)/(cos beta) + (cos beta)/(sin beta) (1); = (sin ^2 beta + cos ^2 beta)/(cos beta sin beta) Algebra; = (1)/(cos beta sin beta) (2); = (1)/(cos beta) × (1)/(sin beta) Algebra; = sec beta csc beta (3); [3pt] Choose from: Pythagorean identity; Quotient identity; Reciprocal identity.

(1) Quotient identity; (2) Pythagorean identity; (3) Reciprocal identity

Give the reason for each numbered step.; [3pt] (1)/(1 - sin beta) + (1)/(1 + sin beta); = (2)/(1 - sin ^2 beta) Algebra; = (2)/(cos ^2 beta) (1); = 2sec ^2 beta (2); [3pt] Choose from: Pythagorean identity; Quotient identity; Reciprocal identity.

(1) Pythagorean identity; (2) Reciprocal identity

Give the reason for each numbered step.; [3pt] sec x - cos x; = (1)/(cos x) - cos x (1); = (1 - cos ^2 x)/(cos x) Algebra; = (sin ^2 x)/(cos x) (2); = sin x × (sin x)/(cos x) Algebra; = sin x tan x (3); [3pt] Choose from: Pythagorean identity; Quotient identity; Reciprocal identity.

(1) Reciprocal identity; (2) Pythagorean identity; (3) Quotient identity

Prove: sec ^2 alpha - tan ^2 alpha = 1

sec ^2 alpha - tan ^2 alpha; = (1 + tan ^2 alpha) - tan ^2 alpha Pythagorean identity; = 1 Algebra

Worksheets · Precalculus

Proving trig identities worksheet

Work on one side only, usually the more complicated one, and change it one step at a time until it is the other side. Each step is either an identity or plain algebra. When stuck, rewrite everything in sines and cosines: it is rarely the shortest proof, but it almost always gets there.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Proving Trig Identities Date Period Prove each identity, or give the reason for each numbered step. Prove the identity Prove: $(\sec θ - 1) (\sec θ + 1) = \tan^{2} θ$ Prove: $\frac{1 + \tan x}{\sec x} = \cos x + \sin x$ Prove: $\csc x - \sin x = \cos x \cot x$ Prove: ${(\sin θ + \cos θ)}^{2} = 1 + 2 \sin θ \cos θ$ Give the reason for each step Give the reason for each numbered step. $\begin{aligned} \tan β + \cot β \\ = \frac{\sin β}{\cos β} + \frac{\cos β}{\sin β} & (1) \\ = \frac{\sin^{2} β + \cos^{2} β}{\cos β \sin β} & Algebra \\ = \frac{1}{\cos β \sin β} & (2) \\ = \frac{1}{\cos β} \cdot \frac{1}{\sin β} & Algebra \\ = \sec β \csc β & (3) \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity. Give the reason for each numbered step. $\begin{aligned} \frac{\cos β}{1 - \sin β} \\ = \frac{\cos β (1 + \sin β)}{1 - \sin^{2} β} & Algebra \\ = \frac{\cos β (1 + \sin β)}{\cos^{2} β} & (1) \\ = \frac{1 + \sin β}{\cos β} & Algebra \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity. Give the reason for each numbered step. $\begin{aligned} \frac{1}{1 - \sin β} + \frac{1}{1 + \sin β} \\ = \frac{2}{1 - \sin^{2} β} & Algebra \\ = \frac{2}{\cos^{2} β} & (1) \\ = 2 \sec^{2} β & (2) \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity. Give the reason for each numbered step. $\begin{aligned} \sec x - \cos x \\ = \frac{1}{\cos x} - \cos x & (1) \\ = \frac{1 - \cos^{2} x}{\cos x} & Algebra \\ = \frac{\sin^{2} x}{\cos x} & (2) \\ = \sin x \cdot \frac{\sin x}{\cos x} & Algebra \\ = \sin x \tan x & (3) \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Proving Trig Identities

Date Period

Prove each identity, or give the reason for each numbered step.

Prove the identity
Prove:
$(\sec θ - 1) (\sec θ + 1) = \tan^{2} θ$
Prove:
$\frac{1 + \tan x}{\sec x} = \cos x + \sin x$
Prove:
$\csc x - \sin x = \cos x \cot x$
Prove:
${(\sin θ + \cos θ)}^{2} = 1 + 2 \sin θ \cos θ$

Give the reason for each step
Give the reason for each numbered step. $\begin{aligned} \tan β + \cot β \\ = \frac{\sin β}{\cos β} + \frac{\cos β}{\sin β} & (1) \\ = \frac{\sin^{2} β + \cos^{2} β}{\cos β \sin β} & Algebra \\ = \frac{1}{\cos β \sin β} & (2) \\ = \frac{1}{\cos β} \cdot \frac{1}{\sin β} & Algebra \\ = \sec β \csc β & (3) \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity.
Give the reason for each numbered step. $\begin{aligned} \frac{\cos β}{1 - \sin β} \\ = \frac{\cos β (1 + \sin β)}{1 - \sin^{2} β} & Algebra \\ = \frac{\cos β (1 + \sin β)}{\cos^{2} β} & (1) \\ = \frac{1 + \sin β}{\cos β} & Algebra \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity.
Give the reason for each numbered step. $\begin{aligned} \frac{1}{1 - \sin β} + \frac{1}{1 + \sin β} \\ = \frac{2}{1 - \sin^{2} β} & Algebra \\ = \frac{2}{\cos^{2} β} & (1) \\ = 2 \sec^{2} β & (2) \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity.
Give the reason for each numbered step. $\begin{aligned} \sec x - \cos x \\ = \frac{1}{\cos x} - \cos x & (1) \\ = \frac{1 - \cos^{2} x}{\cos x} & Algebra \\ = \frac{\sin^{2} x}{\cos x} & (2) \\ = \sin x \cdot \frac{\sin x}{\cos x} & Algebra \\ = \sin x \tan x & (3) \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity.



Sigma Prep · sigmaprep.io/worksheets Name Proving Trig IdentitiesAnswer keyVersion 1 Date Period Prove each identity, or give the reason for each numbered step. Prove the identity Prove: $(\sec θ - 1) (\sec θ + 1) = \tan^{2} θ$ $\begin{aligned} (\sec θ - 1) (\sec θ + 1) \\ = \sec^{2} θ - 1 & Algebra \\ = \tan^{2} θ & Pythagorean identity \end{aligned}$ Prove: $\frac{1 + \tan x}{\sec x} = \cos x + \sin x$ $\begin{aligned} \frac{1 + \tan x}{\sec x} \\ = \cos x (1 + \tan x) & Reciprocal identity \\ = \cos x + \cos x \tan x & Algebra \\ = \cos x + \cos x \cdot \frac{\sin x}{\cos x} & Quotient identity \\ = \cos x + \sin x & Algebra \end{aligned}$ Prove: $\csc x - \sin x = \cos x \cot x$ $\begin{aligned} \csc x - \sin x \\ = \frac{1}{\sin x} - \sin x & Reciprocal identity \\ = \frac{1 - \sin^{2} x}{\sin x} & Algebra \\ = \frac{\cos^{2} x}{\sin x} & Pythagorean identity \\ = \cos x \cdot \frac{\cos x}{\sin x} & Algebra \\ = \cos x \cot x & Quotient identity \end{aligned}$ Prove: ${(\sin θ + \cos θ)}^{2} = 1 + 2 \sin θ \cos θ$ $\begin{aligned} {(\sin θ + \cos θ)}^{2} \\ = \sin^{2} θ + 2 \sin θ \cos θ + \cos^{2} θ & Algebra \\ = 1 + 2 \sin θ \cos θ & Pythagorean identity \end{aligned}$ Give the reason for each step Give the reason for each numbered step. $\begin{aligned} \tan β + \cot β \\ = \frac{\sin β}{\cos β} + \frac{\cos β}{\sin β} & (1) \\ = \frac{\sin^{2} β + \cos^{2} β}{\cos β \sin β} & Algebra \\ = \frac{1}{\cos β \sin β} & (2) \\ = \frac{1}{\cos β} \cdot \frac{1}{\sin β} & Algebra \\ = \sec β \csc β & (3) \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity. $\begin{aligned} (1) Quotient identity \\ (2) Pythagorean identity \\ (3) Reciprocal identity \end{aligned}$ Give the reason for each numbered step. $\begin{aligned} \frac{\cos β}{1 - \sin β} \\ = \frac{\cos β (1 + \sin β)}{1 - \sin^{2} β} & Algebra \\ = \frac{\cos β (1 + \sin β)}{\cos^{2} β} & (1) \\ = \frac{1 + \sin β}{\cos β} & Algebra \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity. $\begin{aligned} (1) Pythagorean identity \end{aligned}$ Give the reason for each numbered step. $\begin{aligned} \frac{1}{1 - \sin β} + \frac{1}{1 + \sin β} \\ = \frac{2}{1 - \sin^{2} β} & Algebra \\ = \frac{2}{\cos^{2} β} & (1) \\ = 2 \sec^{2} β & (2) \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity. $\begin{aligned} (1) Pythagorean identity \\ (2) Reciprocal identity \end{aligned}$ Give the reason for each numbered step. $\begin{aligned} \sec x - \cos x \\ = \frac{1}{\cos x} - \cos x & (1) \\ = \frac{1 - \cos^{2} x}{\cos x} & Algebra \\ = \frac{\sin^{2} x}{\cos x} & (2) \\ = \sin x \cdot \frac{\sin x}{\cos x} & Algebra \\ = \sin x \tan x & (3) \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity. $\begin{aligned} (1) Reciprocal identity \\ (2) Pythagorean identity \\ (3) Quotient identity \end{aligned}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Proving Trig IdentitiesAnswer keyVersion 1

Date Period

Prove each identity, or give the reason for each numbered step.

Prove the identity
Prove:
$(\sec θ - 1) (\sec θ + 1) = \tan^{2} θ$ $\begin{aligned} (\sec θ - 1) (\sec θ + 1) \\ = \sec^{2} θ - 1 & Algebra \\ = \tan^{2} θ & Pythagorean identity \end{aligned}$
Prove:
$\frac{1 + \tan x}{\sec x} = \cos x + \sin x$ $\begin{aligned} \frac{1 + \tan x}{\sec x} \\ = \cos x (1 + \tan x) & Reciprocal identity \\ = \cos x + \cos x \tan x & Algebra \\ = \cos x + \cos x \cdot \frac{\sin x}{\cos x} & Quotient identity \\ = \cos x + \sin x & Algebra \end{aligned}$
Prove:
$\csc x - \sin x = \cos x \cot x$ $\begin{aligned} \csc x - \sin x \\ = \frac{1}{\sin x} - \sin x & Reciprocal identity \\ = \frac{1 - \sin^{2} x}{\sin x} & Algebra \\ = \frac{\cos^{2} x}{\sin x} & Pythagorean identity \\ = \cos x \cdot \frac{\cos x}{\sin x} & Algebra \\ = \cos x \cot x & Quotient identity \end{aligned}$
Prove:
${(\sin θ + \cos θ)}^{2} = 1 + 2 \sin θ \cos θ$ $\begin{aligned} {(\sin θ + \cos θ)}^{2} \\ = \sin^{2} θ + 2 \sin θ \cos θ + \cos^{2} θ & Algebra \\ = 1 + 2 \sin θ \cos θ & Pythagorean identity \end{aligned}$

Give the reason for each step
Give the reason for each numbered step. $\begin{aligned} \tan β + \cot β \\ = \frac{\sin β}{\cos β} + \frac{\cos β}{\sin β} & (1) \\ = \frac{\sin^{2} β + \cos^{2} β}{\cos β \sin β} & Algebra \\ = \frac{1}{\cos β \sin β} & (2) \\ = \frac{1}{\cos β} \cdot \frac{1}{\sin β} & Algebra \\ = \sec β \csc β & (3) \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity. $\begin{aligned} (1) Quotient identity \\ (2) Pythagorean identity \\ (3) Reciprocal identity \end{aligned}$
Give the reason for each numbered step. $\begin{aligned} \frac{\cos β}{1 - \sin β} \\ = \frac{\cos β (1 + \sin β)}{1 - \sin^{2} β} & Algebra \\ = \frac{\cos β (1 + \sin β)}{\cos^{2} β} & (1) \\ = \frac{1 + \sin β}{\cos β} & Algebra \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity. $\begin{aligned} (1) Pythagorean identity \end{aligned}$
Give the reason for each numbered step. $\begin{aligned} \frac{1}{1 - \sin β} + \frac{1}{1 + \sin β} \\ = \frac{2}{1 - \sin^{2} β} & Algebra \\ = \frac{2}{\cos^{2} β} & (1) \\ = 2 \sec^{2} β & (2) \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity. $\begin{aligned} (1) Pythagorean identity \\ (2) Reciprocal identity \end{aligned}$
Give the reason for each numbered step. $\begin{aligned} \sec x - \cos x \\ = \frac{1}{\cos x} - \cos x & (1) \\ = \frac{1 - \cos^{2} x}{\cos x} & Algebra \\ = \frac{\sin^{2} x}{\cos x} & (2) \\ = \sin x \cdot \frac{\sin x}{\cos x} & Algebra \\ = \sin x \tan x & (3) \end{aligned}$ Choose from: Pythagorean identity; Quotient identity; Reciprocal identity. $\begin{aligned} (1) Reciprocal identity \\ (2) Pythagorean identity \\ (3) Quotient identity \end{aligned}$

How to do these

Prove:
$\tan θ + \cot θ = \sec θ \csc θ$

Quotient identities: the left side is sin/cos + cos/sin.
Common denominator: (sin squared + cos squared) over cos times sin.
Pythagorean identity: the top is 1, so it is 1 over cos times sin.
Reciprocal identities: that is sec times csc.

The answer is $\tan θ + \cot θ = \sec θ \csc θ$ .

Where students go wrong

Working on both sides at once, as if solving an equation. Adding or multiplying both sides assumes the two are equal, which is the thing being proved. Change one side until it matches the other.

Also called verifying trig identities, trig identity proofs, prove the identity or verify the identity.

What you can put on this worksheet

Prove the identity: Prove: $\sec^{2} α - \tan^{2} α = 1$
→ Answer
$\begin{aligned} \sec^{2} α - \tan^{2} α \\ = (1 + \tan^{2} α) - \tan^{2} α & Pythagorean identity \\ = 1 & Algebra \end{aligned}$
Give the reason for each step: Give the reason for each numbered step.
$\begin{aligned} \sec^{2} α - \tan^{2} α \\ = (1 + \tan^{2} α) - \tan^{2} α & (1) \\ = 1 & Algebra \end{aligned}$
Choose from: Pythagorean identity; Quotient identity; Reciprocal identity.
→ Answer
$(1)$ Pythagorean identity

Questions about these worksheets

Yes. Every proving trig identities sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the proving trig identities questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 2: prove the identity and give the reason for each step. Tick as many as you want and set how many of each, or let it spread them evenly.

Working on both sides at once, as if solving an equation. Adding or multiplying both sides assumes the two are equal, which is the thing being proved. Change one side until it matches the other.

Proving trig identities worksheet

Select your difficulty

Prove the identity

Give the reason for each step

Prove the identity

Give the reason for each step

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these proving trig identities worksheets free?+

Do the proving trig identities worksheets come with an answer key?+

Are these proving trig identities worksheets printable?+

What grade are these proving trig identities worksheets for?+

Can I get a different version of the same proving trig identities worksheet?+

Is this a free alternative to Kuta Software for proving trig identities?+

Can I choose how hard the proving trig identities questions are?+

What kinds of question are on a proving trig identities worksheet?+

What do students get wrong with proving trig identities?+