cos⁴ A/2 - sin⁴ A/2 = cos A | Prove the trigonometric identity | SciPiPupil

Question: Prove the following trigonometric identity: $cos⁴\dfrac{A}{2} - sin⁴\dfrac{A}{2}$ = cos A.

Solution:

LHS

= $cos⁴\dfrac{A}{2} - sin⁴\dfrac{A}{2}$

= $(cos²\dfrac{A}{2})² - (sin²\dfrac{A}{2})²$

= $(cos²\dfrac{A}{2} - sin²\dfrac{A}{2}) * (cos²\dfrac{A}{2} + sin²\dfrac{A}{2})$

= $1 * cos(2*\dfrac{1}{2})$

= $cos A$

= RHS