Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

After replacing the sine of a negative angle with its equivalent, simplifying, and applying the Pythagorean identity, we have confirmed that the left-hand side of the equation is equivalent to the right-hand side. Therefore, the given identity is verified.

Step 1: Rewriting the term

Notice that \(\sin(-y) = - \sin y \). Therefore, rewrite the problem as: (1 + \sin y)(1 - \sin y)

Step 2: Simplifying the expression

Simplify the above expression using the difference of squares identity, which states that (a + b)(a - b) equals \(a^2 - b^2\). Therefore, the expression simplifies to \(1^2 - \sin^2 y\).

Step 3: Using the Pythagorean Identity

Now use the Pythagorean Identity, which says that \(\sin^2(x) + \cos^2(x) = 1 \) or re-arranged \(1 - \sin^2(x) = \cos^2(x)\). So our expression \(1^2 - \sin^2 y\) becomes \(\cos^2 y\) after applying the Pythagorean Identity. This matches the right hand side of the original equation, and thus, the identity is verified.