{ \cos^(4)\alpha} \: + \sin^(2) \alpha \: \: = \: (1)/(4) (3 + \cos4 \alpha ) Prove::​

Question

`{ \cos^(4)\alpha} \: + \sin^(2) \alpha \: \: = \: (1)/(4) (3 + \cos4 \alpha )`

Prove::​

Answer 1

Proved

Explanation:

`cos^4\alpha +sin^4\alpha =(1)/(4)(3+cos4\alpha )\\\\`

Take the Left Hand Side:

`cos^4\alpha +sin^4\alpha\\\\(cos^2\alpha )^2+(sin^2\alpha )^2\\\\((1+cos2\alpha )/(2))^2+((1-cos2\alpha )/(2))^2 \\\\(1+2cos2\alpha +cos^22\alpha )/(4)+(1-2cos2\alpha +cos^22\alpha )/(4) \\\\`

`(1+2cos2\alpha +cos^22\alpha +1-2cos2\alpha +cos^22\alpha )/(4) \\\\(2+2cos^22\alpha )/(4) \\\\(1+cos^22\alpha )/(2) \\\\(1)/(2)(1+cos^22\alpha )\\\\`

`(1)/(2)(1+(1+cos4\alpha )/(2))`

`(1)/(2)((2+1+cos4\alpha )/(2))\\\\(1)/(4)(3+cos4\alpha )\\\\`

Hence Proved!

The following identities were used are attached in an image