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Chapter 5: Problem 37

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

Short Answer

Expert verified
The given identity \((1+\sin y)(1+\sin (-y))=\cos^{2} y\) is verified, as we have shown the steps transforming left hand side to \(\cos^{2} y\), which is the right hand side.

Step by step solution

01

Apply the sine property

Replace \(\sin(-y)\) with \(-\sin y\). So, the left hand side (LHS) of the equation becomes \((1+\sin y)(1-\sin y)\).
02

Multiply the expressions

Use the multiplication formula (a+b)(a-b) = a^2 - b^2 to multiply out the LHS. This results in \(1 - (\sin y)^2\).
03

Apply the Pythagorean trigonometric identity

Recall and apply the identity \(\cos^{2} y = 1 - \sin^{2} y\). So, the LHS, \(1 - (\sin y)^2\), simplifies to \(\cos^{2} y\).

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