Problem 57 - Precalculus Essentials

Question

Verify each identity. $$(\cos \theta-\sin \theta)^{2}+(\cos \theta+\sin \theta)^{2}=2$$

Step-by-Step Solution

Answer

The identity is verified, as after applying the Pythagorean identity and simplifying, we get \(2=2\) which is true.

Step 1: Expand

Expand each term. For \((\cos \theta-\sin \theta)^{2}\), calculate \(cos^2 \theta -2 \cos \theta \sin \theta+ sin ^2 \theta\). And for \((\cos \theta+\sin \theta)^{2}\), calculate \(cos^2 \theta + 2 \cos \theta \sin \theta + sin^2 \theta\).

Step 2: Simplify

Simplify the expressions by adding the two results together. The \(-2 \cos \theta \sin \theta\) term and the \(+2 \cos \theta \sin \theta\) term will cancel out each other, resulting in \(2 cos^2 \theta + 2 sin^2 \theta\).

Step 3: Apply Trig Identity

We know from the Pythagorean identity that \(cos^2 \theta + sin^2 \theta = 1\). Substituting this into our expression, we get \(2 * 1 = 2\).