Skill: Apply Pythagorean identities

Explanation and Free Practice Resources

Pythagorean identities express the fundamental relationships among real-valued trigonometric functions: $\sin^2\theta+\cos^2\theta=1$ , $1+\tan^2\theta=\sec^2\theta$ , and $1+\cot^2\theta=\csc^2\theta$ . The learner applies these identities to rewrite expressions, find missing trigonometric values, establish equivalent forms, and simplify equations while respecting signs, quadrants, and the domains where $\tan$ , $\cot$ , $\sec$ , or $\csc$ are defined; complex-valued, hyperbolic, and more advanced generalized identities are outside this scope.

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Detailed Explanation: Apply Pythagorean identities

Pythagorean identities help you replace one trigonometric expression with an equivalent one:

\sin^2\theta+\cos^2\theta=1

1+\tan^2\theta=\sec^2\theta

1+\cot^2\theta=\csc^2\theta

Choose the identity that contains the function you know. Then use the quadrant to select the correct sign.

Worked example

Suppose $\theta$ is in Quadrant II and

\tan\theta=-\frac{3}{4}.

Find $\sin\theta$ and $\cos\theta$ .

Because tangent is given, use

1+\tan^2\theta=\sec^2\theta.

Substitute $\tan\theta=-\frac34$ :

1+\left(-\frac34\right)^2=\sec^2\theta

1+\frac{9}{16}=\sec^2\theta

\sec^2\theta=\frac{25}{16}.

Taking square roots gives

\sec\theta=\pm\frac54.

In Quadrant II, cosine is negative. Since $\sec\theta=\frac{1}{\cos\theta}$ , secant is also negative:

\sec\theta=-\frac54.

Therefore,

\cos\theta=\frac{1}{\sec\theta} =\frac{1}{-\frac54} =-\frac45.

Now use $\tan\theta=\frac{\sin\theta}{\cos\theta}$ :

-\frac34=\frac{\sin\theta}{-\frac45}.

Multiply by $-\frac45$ :

\sin\theta=\left(-\frac34\right)\left(-\frac45\right)=\frac35.

Thus,

\boxed{\sin\theta=\frac35,\qquad \cos\theta=-\frac45}.

The signs match Quadrant II: sine is positive and cosine is negative.

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Trigonometry Identities - General Substitution Brackets Times Identity (without Identity)

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