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

Related Notes And Solutions:

Link: Introduction To Trigonometry
Link: Values of Trigonometric Ratios
Link: Compound Angles

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