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Question: The value of \(6(\sin^{6}\theta + \cos^{6}\theta) - 9(\sin^{4}\theta + \cos^{4}\theta) + 4\) is...

The value of 6(sin⁡6θ+cos⁡6θ)−9(sin⁡4θ+cos⁡4θ)+46(\sin^{6}\theta + \cos^{6}\theta) - 9(\sin^{4}\theta + \cos^{4}\theta) + 4 is

A

–3

B

0

C

1

D

3

Answer

1

Explanation

Solution

6(sin⁡6θ+cos⁡6θ)−9(sin⁡4θ+cos⁡4θ)+46(\sin^{6}\theta + \cos^{6}\theta) - 9(\sin^{4}\theta + \cos^{4}\theta) + 4

=6[(sin⁡2θ+cos⁡2θ)3−3sin⁡2θcos⁡2θ(sin⁡2θ+cos⁡2θ)]−9[(sin⁡2θ+cos⁡2θ)2−2sin⁡2θcos⁡2]+4= 6\lbrack(\sin^{2}\theta + \cos^{2}\theta)^{3} - 3\sin^{2}\theta\cos^{2}\theta(\sin^{2}\theta + \cos^{2}\theta)\rbrack - 9\lbrack(\sin^{2}\theta + \cos^{2}\theta)^{2} - 2\sin^{2}\theta\cos^{2}\rbrack + 4

=6[1−3sin⁡2θcos⁡2θ]−9[1−2sin⁡2θcos⁡2θ]+4=6−9+4=1= 6\lbrack 1 - 3\sin^{2}\theta\cos^{2}\theta\rbrack - 9\lbrack 1 - 2\sin^{2}\theta\cos^{2}\theta\rbrack + 4 = 6 - 9 + 4 = 1.