Prove the following identities. (These two identities, along...

Prove the following identities. (These two identities, along with $\sin ^{2} \theta+\cos ^{2} \theta=1,$ are known as the Pythagorean identities. They will be discussed in the next chapter.)
(a) $\tan ^{2} \theta=\sec ^{2} \theta-1$
(b) $\cot ^{2} \theta=\csc ^{2} \theta-1$

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Key Concepts

Pythagorean Identities

Pythagorean identities are fundamental relationships between the trigonometric functions derived from the Pythagorean theorem. They typically relate the squares of sine, cosine, tangent, cotangent, secant, and cosecant to each other, and serve as key tools for proving and deriving additional trigonometric identities.

Trigonometric Function Definitions

Understanding the definitions of trigonometric functions in terms of sine and cosine is crucial. For example, tangent is defined as sine divided by cosine, secant as the reciprocal of cosine, cotangent as cosine divided by sine, and cosecant as the reciprocal of sine. These definitions allow one to express more complex identities in terms of sine and cosine, facilitating their proof.

Algebraic Manipulation in Trigonometry

Algebraic manipulation plays a vital role in proving trigonometric identities. This includes techniques such as rewriting functions in alternative forms, factoring, and cross-multiplication. By expressing trigonometric functions in terms of sine and cosine and applying algebraic operations, more complex identities can be derived from the fundamental Pythagorean identity.

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