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Chapter 5: Problem 37

Verify the identity. $$(1+\sin y)[1+\sin (-y)]=\cos ^{2} y$$

Short Answer

Expert verified
By using the properties of the sine function and the Pythagorean identity, we're able to transform \((1+\sin y)[1+\sin (-y)]\) into \(\cos^{2} y\). Therefore, the identity is verified.

Step by step solution

01

- Expand The Left-hand side

To examine the equation more clearly, start with the expansion of the left-hand side: \((1+\sin y)[1+\sin (-y)]\) using distributive law. Since \(\sin (-y) = -\sin y\), the expression changes to \((1+\sin y)(1 - \sin y )\).
02

- Calculate Result

Implementing the formula \(a^2 - b^2 = (a-b)(a+b)\) in our expression, we get \(1 - \sin^2 y\).
03

- Recognize the Pythagorean Identity

We know from the Pythagorean identity in trigonometry that \(\sin^2 y + \cos^2 y = 1\). Consequently, we can replace \(1- \sin^2 y\) with \(\cos^2 y\).

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