Problem 37 - Precalculus Essentials

Question

Verify each identity. $$\frac{\sin ^{2} x-\cos ^{2} x}{\sin x+\cos x}=\sin x-\cos x$$

Step-by-Step Solution

Answer

After simplifying the left-hand side of the identity to \( \sin x - \cos x \), both sides of the equation are equal, proving the identity

Step 1: Rewrite numerator using difference of squares

The numerator \( \sin^{2}x - \cos^{2}x \) on the left-hand side of the given identity can be written as a difference of squares: \( (\sin x + \cos x)(\sin x - \cos x) \)

Step 2: Simplify the left-hand side

Since the denominator of the left-hand side of the identity is \( \sin x + \cos x \), we can divide it by the same factor from the numerator, which leaves us with \( \sin x - \cos x \)

Step 3: Compare

Now both sides of the identity are the same: \( \sin x - \cos x = \sin x - \cos x \). This confirms that the given expression is indeed an identity