Use the first Pythagorean identity sin^2 θ+cos^2 θ=1 to...

Use the first Pythagorean identity $\sin ^{2} \theta+\cos ^{2} \theta=1$ to prove the second and third Pythagorean identities listed in Theorem 0.26. (Hint: To prove the second identity, divide both sides of the first identity by $\cos ^{2} x$. A similar strategy will prove the third identity.)

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Key Concepts

Pythagorean Identity

The Pythagorean identity, sin²? + cos²? = 1, is a foundational relation in trigonometry that connects the sine and cosine functions for any angle. It is pivotal because it forms the basis for deriving other identities by allowing substitutions and algebraic rearrangements that reveal relationships among different trigonometric functions.

Trigonometric Ratios

Understanding the definitions of trigonometric ratios such as tangent, cotangent, secant, and cosecant in terms of sine and cosine is essential. For instance, recognizing that tan? = sin?/cos? and sec? = 1/cos? provides the means to manipulate and transform the Pythagorean identity into other useful forms.

Algebraic Manipulation

Algebraic techniques, particularly division by a nonzero function and rearrangement of terms, are important for transforming identities. Dividing the Pythagorean identity by the square of a trigonometric function, such as cos²? or sin²?, facilitates the derivation of related identities by isolating and reinterpreting the remaining expressions in terms of well-known trigonometric ratios.

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