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
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