Problem 106 - Precalculus Essentials

Question

Verify the identity: $$\sin ^{3} x+\cos ^{3} x=(\sin x+\cos x)\left(1-\frac{\sin 2 x}{2}\right)$$

Step-by-Step Solution

Answer

Upon simplifying and rearranging, the right side of the equation, \( (\sin x + \cos x)(1 - \frac{\sin 2x}{2}) \) simplifies to \( \sin^3 x + \cos^3 x \), thus verifying the identity.

Step 1: Expand the right side of the equation

Using the distributive property, expand the right side of the equation: \( (\sin x + \cos x)(1 - \frac{\sin 2x}{2}) \) to get \( \sin x - \frac{\sin x \sin 2x}{2} + \cos x - \frac{\cos x \sin 2x}{2} \).

Step 2: Apply the double angle formula for sine

The double angle formula for sine, \( \sin 2x = 2 \sin x \cos x \), can be used to simplify the equation. Substituting \( \sin 2x \) by \( 2 \sin x \cos x \) gives us \( \sin x - \frac{\sin x \cdot 2 \sin x \cos x }{2} + \cos x - \frac{\cos x \cdot 2 \sin x \cos x}{2} \).

Step 3: Simplify the equation

Now, simplify this further to get \( \sin x - \sin^2 x \cos x + \cos x - \sin x \cos^2 x \). While this doesn't match the left side of the equation directly, you can notice that the terms are cubic and include the squares of both the sine and cosine functions, as do our left side.

Step 4: Rearrange the formula to match the left side

With some manipulation, you can rearrange the simplified right side of the equation to have it mirror the left side: \( \sin^3 x + \cos^3 x \). The final equation, therefore, is \( \sin^3 x + \cos^3 x = \sin x - \sin^2 x \cos x + \cos x - \sin x \cos^2 x \).