Some Applications of Trigonometry

Prove the following identity: 2 (sin 6 θ + cos 6 θ ) -3 (sin 4 θ + cos 4 θ ) + 1 = 0 - Wired Faculty

Question

Prove the following identity:
2 (sin⁶θ + cos⁶θ) -3 (sin⁴θ + cos⁴θ) + 1 = 0

Solution

L.H..S. = $space 2 left parenthesis sin to the power of 6 straight theta plus cos to the power of 6 straight theta right parenthesis space minus space 3 left parenthesis sin to the power of 4 straight theta plus cos to the power of 4 straight theta right parenthesis plus 1$
$equals 2 left square bracket left parenthesis sin squared straight theta right parenthesis cubed plus left parenthesis cos squared straight theta right parenthesis cubed right square bracket minus 3 left square bracket left parenthesis sin squared straight theta right parenthesis squared plus left parenthesis cos squared straight theta right parenthesis squared right square bracket plus 1 equals 2 left square bracket left parenthesis sin squared straight theta plus cos squared straight theta right parenthesis space minus space 3 space sin squared straight theta. space cos squared straight theta right square bracket space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus 3 left square bracket left parenthesis sin squared straight theta plus cos squared straight theta right square bracket squared equals space 2 left square bracket left parenthesis 1 right parenthesis cubed minus 3 sin squared straight theta. space cos squared straight theta right square bracket space minus 3 space left square bracket 1 minus 2 sin squared cos squared straight theta right square bracket plus 1 equals 2 space left parenthesis 1 minus 3 sin squared straight theta. space cos squared straight theta right parenthesis space minus 3 space left parenthesis 1 minus 2 sin squared cos squared straight theta right square bracket plus 1$

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