sin theta = 4sqrt(17)17 and cos theta > 0. Find the other five ratios.

cos theta = sqrt(17)17 tan theta = 4; [6pt] csc theta = sqrt(17)4 sec theta = sqrt(17); [6pt] cot theta = 14

Fundamental trig identities worksheet for Precalculus

Four families do almost all the work. Reciprocal: csc is 1 over sin, sec is 1 over cos, cot is 1 over tan. Quotient: tan is sin over cos. Pythagorean: sin squared plus cos squared is 1, and dividing that through gives the other two. Odd-even: cos(-x) is cos x, while sin and tan change sign.

Also taught in Algebra 2, at its own level and with its own examples.

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 Fundamental Trigonometric Identities Date Period Simplify each expression, or find the value asked for. Simplify with the fundamental identities Simplify: $(1 - \sin θ) (1 + \sin θ)$ Simplify: $\cos^{2} θ (1 + \tan^{2} θ)$ Simplify: $\frac{1 - \cos^{2} β}{\sin β}$ Simplify: $\frac{\sec x}{\csc x}$ Simplify: $\csc θ \tan θ$ Simplify: $\sec β \cot β$ Simplify: $\frac{\csc^{2} β - 1}{\csc^{2} β}$ Simplify: $\sin^{2} θ \csc θ$ Simplify: $\sin θ \sec θ \cot θ$ Simplify: $\tan x \csc x \cos x$ Find a ratio from another ratio and the quadrant If $\sin θ = - \frac{24}{25}$ and $θ$ is in quadrant IV, find $\tan θ$ If $\cos θ = \frac{15}{17}$ and $θ$ is in quadrant I, find $\sin θ$ If $\cos θ = \frac{15}{17}$ and $θ$ is in quadrant I, find $\tan θ$ If $\cos θ = - \frac{12}{13}$ and $θ$ is in quadrant III, find $\sin θ$ If $\cos θ = \frac{35}{37}$ and $θ$ is in quadrant IV, find $\sec θ$ If $\sin θ = \frac{9}{41}$ and $θ$ is in quadrant I, find $\sec θ$ If $\tan θ = \frac{15}{8}$ and $θ$ is in quadrant I, find $\csc θ$ If $\tan θ = - \frac{20}{21}$ and $θ$ is in quadrant IV, find $\cos θ$ If $\cos θ = - \frac{5}{13}$ and $θ$ is in quadrant III, find $\sec θ$ If $\cos θ = - \frac{21}{29}$ and $θ$ is in quadrant III, find $\csc θ$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Fundamental Trigonometric Identities

Date Period

Simplify each expression, or find the value asked for.

Simplify with the fundamental identities
Simplify:
$(1 - \sin θ) (1 + \sin θ)$
Simplify:
$\cos^{2} θ (1 + \tan^{2} θ)$
Simplify:
$\frac{1 - \cos^{2} β}{\sin β}$
Simplify:
$\frac{\sec x}{\csc x}$
Simplify: $\csc θ \tan θ$
Simplify: $\sec β \cot β$
Simplify:
$\frac{\csc^{2} β - 1}{\csc^{2} β}$
Simplify: $\sin^{2} θ \csc θ$
Simplify: $\sin θ \sec θ \cot θ$
Simplify: $\tan x \csc x \cos x$

Find a ratio from another ratio and the quadrant
If $\sin θ = - \frac{24}{25}$ and $θ$ is in quadrant IV, find $\tan θ$
If $\cos θ = \frac{15}{17}$ and $θ$ is in quadrant I, find $\sin θ$
If $\cos θ = \frac{15}{17}$ and $θ$ is in quadrant I, find $\tan θ$
If $\cos θ = - \frac{12}{13}$ and $θ$ is in quadrant III, find $\sin θ$
If $\cos θ = \frac{35}{37}$ and $θ$ is in quadrant IV, find $\sec θ$
If $\sin θ = \frac{9}{41}$ and $θ$ is in quadrant I, find $\sec θ$
If $\tan θ = \frac{15}{8}$ and $θ$ is in quadrant I, find $\csc θ$
If $\tan θ = - \frac{20}{21}$ and $θ$ is in quadrant IV, find $\cos θ$
If $\cos θ = - \frac{5}{13}$ and $θ$ is in quadrant III, find $\sec θ$
If $\cos θ = - \frac{21}{29}$ and $θ$ is in quadrant III, find $\csc θ$



Sigma Prep · sigmaprep.io/worksheets Name Fundamental Trigonometric IdentitiesAnswer keyVersion 1 Date Period Simplify each expression, or find the value asked for. Simplify with the fundamental identities Simplify: $(1 - \sin θ) (1 + \sin θ)$ $\cos^{2} θ$ Simplify: $\cos^{2} θ (1 + \tan^{2} θ)$ $1$ Simplify: $\frac{1 - \cos^{2} β}{\sin β}$ $\sin β$ Simplify: $\frac{\sec x}{\csc x}$ $\tan x$ Simplify: $\csc θ \tan θ$ $\sec θ$ Simplify: $\sec β \cot β$ $\csc β$ Simplify: $\frac{\csc^{2} β - 1}{\csc^{2} β}$ $\cos^{2} β$ Simplify: $\sin^{2} θ \csc θ$ $\sin θ$ Simplify: $\sin θ \sec θ \cot θ$ $1$ Simplify: $\tan x \csc x \cos x$ $1$ Find a ratio from another ratio and the quadrant If $\sin θ = - \frac{24}{25}$ and $θ$ is in quadrant IV, find $\tan θ$ $- \frac{24}{7}$ If $\cos θ = \frac{15}{17}$ and $θ$ is in quadrant I, find $\sin θ$ $\frac{8}{17}$ If $\cos θ = \frac{15}{17}$ and $θ$ is in quadrant I, find $\tan θ$ $\frac{8}{15}$ If $\cos θ = - \frac{12}{13}$ and $θ$ is in quadrant III, find $\sin θ$ $- \frac{5}{13}$ If $\cos θ = \frac{35}{37}$ and $θ$ is in quadrant IV, find $\sec θ$ $\frac{37}{35}$ If $\sin θ = \frac{9}{41}$ and $θ$ is in quadrant I, find $\sec θ$ $\frac{41}{40}$ If $\tan θ = \frac{15}{8}$ and $θ$ is in quadrant I, find $\csc θ$ $\frac{17}{15}$ If $\tan θ = - \frac{20}{21}$ and $θ$ is in quadrant IV, find $\cos θ$ $\frac{21}{29}$ If $\cos θ = - \frac{5}{13}$ and $θ$ is in quadrant III, find $\sec θ$ $- \frac{13}{5}$ If $\cos θ = - \frac{21}{29}$ and $θ$ is in quadrant III, find $\csc θ$ $- \frac{29}{20}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Fundamental Trigonometric IdentitiesAnswer keyVersion 1

Date Period

Simplify each expression, or find the value asked for.

Simplify with the fundamental identities
Simplify:
$(1 - \sin θ) (1 + \sin θ)$ $\cos^{2} θ$
Simplify:
$\cos^{2} θ (1 + \tan^{2} θ)$ $1$
Simplify:
$\frac{1 - \cos^{2} β}{\sin β}$ $\sin β$
Simplify:
$\frac{\sec x}{\csc x}$ $\tan x$
Simplify: $\csc θ \tan θ$ $\sec θ$
Simplify: $\sec β \cot β$ $\csc β$
Simplify:
$\frac{\csc^{2} β - 1}{\csc^{2} β}$ $\cos^{2} β$
Simplify: $\sin^{2} θ \csc θ$ $\sin θ$
Simplify: $\sin θ \sec θ \cot θ$ $1$
Simplify: $\tan x \csc x \cos x$ $1$

Find a ratio from another ratio and the quadrant
If $\sin θ = - \frac{24}{25}$ and $θ$ is in quadrant IV, find $\tan θ$ $- \frac{24}{7}$
If $\cos θ = \frac{15}{17}$ and $θ$ is in quadrant I, find $\sin θ$ $\frac{8}{17}$
If $\cos θ = \frac{15}{17}$ and $θ$ is in quadrant I, find $\tan θ$ $\frac{8}{15}$
If $\cos θ = - \frac{12}{13}$ and $θ$ is in quadrant III, find $\sin θ$ $- \frac{5}{13}$
If $\cos θ = \frac{35}{37}$ and $θ$ is in quadrant IV, find $\sec θ$ $\frac{37}{35}$
If $\sin θ = \frac{9}{41}$ and $θ$ is in quadrant I, find $\sec θ$ $\frac{41}{40}$
If $\tan θ = \frac{15}{8}$ and $θ$ is in quadrant I, find $\csc θ$ $\frac{17}{15}$
If $\tan θ = - \frac{20}{21}$ and $θ$ is in quadrant IV, find $\cos θ$ $\frac{21}{29}$
If $\cos θ = - \frac{5}{13}$ and $θ$ is in quadrant III, find $\sec θ$ $- \frac{13}{5}$
If $\cos θ = - \frac{21}{29}$ and $θ$ is in quadrant III, find $\csc θ$ $- \frac{29}{20}$

How to do these

Simplify: $\sec θ \cot θ$

Write both in sine and cosine: (1 / cos) times (cos / sin).
The cosines cancel.
That leaves 1 over sine.

The answer is $\csc θ$ .

Where students go wrong

Writing everything in sine and cosine and then stopping. The answer is usually one function, and 1 over sine should be written as cosecant.

Also called fundamental identities, pythagorean identities, reciprocal identities, quotient identities, even odd identities or simplifying trig expressions.

What you can put on this worksheet

Simplify with the fundamental identities: Simplify: $1 - \cos^{2} α$ → $\sin^{2} α$
One ratio and the quadrant, find another: If $\sin θ = - \frac{15}{17}$ and $θ$ is in quadrant III, find $\cos θ$ → $- \frac{8}{17}$
Cofunction and even-odd values from a decimal: If $\cos t = 0.8$ , find $\csc ⁣ (\frac{π}{2} - t)$ → $1.25$
One ratio and a sign: the others, with radicals: $\sin θ = \frac{4 \sqrt{17}}{17}$ and $\cos θ > 0$ . Find the other five ratios. → $\begin{array}{ll} \cos θ = \frac{\sqrt{17}}{17} & \tan θ = 4 \\ \csc θ = \frac{\sqrt{17}}{4} & \sec θ = \sqrt{17} \\ \cot θ = \frac{1}{4} \end{array}$

Questions about these worksheets

Yes. Every fundamental 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 fundamental 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 4: simplify with the fundamental identities, one ratio and the quadrant, find another, cofunction and even-odd values from a decimal and one ratio and a sign: the others, with radicals. Tick as many as you want and set how many of each, or let it spread them evenly.

Writing everything in sine and cosine and then stopping. The answer is usually one function, and 1 over sine should be written as cosecant.

Fundamental trig identities worksheet for Precalculus

Select your difficulty

Simplify with the fundamental identities

Find a ratio from another ratio and the quadrant

Simplify with the fundamental identities

Find a ratio from another ratio and the quadrant

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these fundamental trig identities worksheets free?+

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

Are these fundamental trig identities worksheets printable?+

What grade are these fundamental trig identities worksheets for?+

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

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

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

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

What do students get wrong with fundamental trig identities?+