Prove that Sin⁶theta/2+ Cos⁶theta/2 = 1/4 ( 1 + 3 cos²theta).
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Step-by-step explanation:
L.H.S. =sin
6
θ+cos
6
θ
=(sin
2
θ)
3
+(cos
2
θ)
3
Put sin
2
θ=a and cos
2
θ=b
∴ L.H.S. =a
3
+b
3
=(a+b)
3
−2ab(a+b)
=(sin
2
θ−cos
2
θ)
3
−3sin
2
θcos
2
θ(sin
2
θ+cos
2
θ)
=(1)
3
−3sin
2
θcos
2
θ[∵sin
2
θ+cos
2
θ=1]
=1−3sin
2
θcos
2
θ
= R.H.S.
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