8.2 Q-8

Exercise 8.2 — Question 8

Problem

Prove the following identity:

${\cos }^{4}x-{\sin }^{4}x=\cos 2x$

Solution

Step 1: Factor as a difference of squares

Recognize that ${\cos }^{4}x-{\sin }^{4}x=({\cos }^{2}x{)}^2-({\sin }^{2}x{)}^2$, which is a difference of two squares:

${\cos }^{4}x-{\sin }^{4}x=({\cos }^{2}x-{\sin }^{2}x)({\cos }^{2}x+{\sin }^{2}x)$

Step 2: Apply the Pythagorean identity

Since ${\cos }^{2}x+{\sin }^{2}x=1$:

$=({\cos }^{2}x-{\sin }^{2}x)(1)$

$={\cos }^{2}x-{\sin }^{2}x$

Step 3: Apply the double-angle identity

Using $\cos 2x={\cos }^{2}x-{\sin }^{2}x$:

$=\cos 2x✓$

---

Key Identities Used