Question

1)

cos^4 = 3/8+1/2cos2x+1/8cos4x

2) Cos^6 theta +sin^6 theta = 1-3/4sin^22theta

Please prove both I will appreciate you

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Answer 1

`\qquad`
`\purple{\twoheadrightarrow\bf 2cos^2A = 1 + cos2A }`

`\qquad`
`\purple{\twoheadrightarrow\bf a^3 +b^3 = (a+b)^3 -3ab(a+b)}`

`\qquad`
`\purple{\twoheadrightarrow\bf 2 sinA cosA = sin2A}`

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1)

`\qquad`
`\twoheadrightarrow\bf L.H.S`

`\qquad`
`\pink{\twoheadrightarrow\bf cos^4 x}`

`\qquad`
`\twoheadrightarrow\sf (1)/(4) (2cos^2x)^2`

`\qquad`
`\twoheadrightarrow\sf (1)/(4)(1+cos2x)^2`

`\qquad`
`\twoheadrightarrow\sf (1)/(4)(1+2cos2x+cos^22x)`

`\qquad`
`\twoheadrightarrow\sf (1)/(4)+(1)/(2)cos2x+(1)/(8)* 2cos^22x`

`\qquad`
`\twoheadrightarrow\sf (1)/(4)+(1)/(2)cos2x+(1)/(8)*(1+cos4x)`

`\qquad`
`\pink{\twoheadrightarrow\bf (1)/(4)+(1)/(2)cos2x+(1)/(8)* cos4x}`

`\qquad`
`\twoheadrightarrow\bf R.H.S`

______________________________________

2)

`\qquad`
`\twoheadrightarrow\bf L.H.S`

`\qquad`
`\pink{\twoheadrightarrow\bf cos^6\theta +sin^6\theta }`

`\qquad`
`\twoheadrightarrow\sf (cos^2\theta)^3 +(sin^2\theta)^3`

`\twoheadrightarrow\sf (cos^2\theta +sin^2\theta)^3 -3cos^2\theta sin^2\theta (cos^2\theta +sin^2\theta)`

`\qquad`
`\twoheadrightarrow\sf 1-3* (1)/(4)* 4(sin\theta cos\theta) ^2`

`\qquad`
`\twoheadrightarrow\sf 1-(3)/(4)(2sin\theta cos\theta) ^2`

`\qquad`
`\pink{\twoheadrightarrow\sf 1-(3)/(4)sin^22\theta}`

`\qquad`
`\twoheadrightarrow\bf R.H.S`