Which of the following identities is true?

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(D) We know that $\cos 5 	heta = \cos (4 	heta + 	heta)$.Using the formula $\cos (A + B) = \cos A \cos B - \sin A \sin B$,we have:$\cos 5 	heta =

Vedclass · Accepted Answer

$\cos 5 	heta = 16 \cos ^5 	heta - 20 \cos ^3 	heta + 5 \cos 	heta$

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(D) We know that $\cos 5 	heta = \cos (4 	heta + 	heta)$.Using the formula $\cos (A + B) = \cos A \cos B - \sin A \sin B$,we have:$\cos 5 	heta = \cos 4 	heta \cos 	heta - \sin 4 	heta \sin 	heta$$= (2 \cos ^2 2 	heta - 1) \cos 	heta - (2 \sin 2 	heta \cos 2 	heta) \sin 	heta$$= \{2(2 \cos ^2 	heta - 1)^2 - 1\} \cos 	heta - 2(2 \sin 	heta \cos 	heta) \cos 2 	heta \sin 	heta$$= \{2(4 \cos ^4 	heta - 4 \cos ^2 	heta + 1) - 1\} \cos 	heta - 4 \sin ^2 	heta \cos 	heta (2 \cos ^2 	heta - 1)$$= (8 \cos ^4 	heta - 8 \cos ^2 	heta + 1) \cos 	heta - 4(1 - \cos ^2 	heta) \cos 	heta (2 \cos ^2 	heta - 1)$$= (8 \cos ^5 	heta - 8 \cos ^3 	heta + \cos 	heta) - 4 \cos 	heta (2 \cos ^2 	heta - 1 - 2 \cos ^4 	heta + \cos ^2 	heta)$$= 8 \cos ^5 	heta - 8 \cos ^3 	heta + \cos 	heta - 4 \cos 	heta (3 \cos ^2 	heta - 1 - 2 \cos ^4 	heta)$$= 8 \cos ^5 	heta - 8 \cos ^3 	heta + \cos 	heta - 12 \cos ^3 	heta + 4 \cos 	heta + 8 \cos ^5 	heta$$= 16 \cos ^5 	heta - 20 \cos ^3 	heta + 5 \cos 	heta$

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