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  4. The possible value of sin 6(θ)+ cos 6(θ)-3 cos 4(θ) is

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Q. The possible value of $\sin ^{6}(\theta)+\cos ^{6}(\theta)-3 \cos ^{4}(\theta)$ is

AP EAMCETAP EAMCET 2020

Solution:

The $\sin ^{6} \theta+\cos ^{6} \theta-3 \cos ^{4} \theta$
$=\left(\sin ^{2} \theta+\cos ^{2} \theta\right)\left(\sin ^{4} \theta+\cos ^{4} \theta-\sin ^{2} \theta \cos ^{2} \theta\right)-3 \cos ^{4} \theta$
$=1 \times\left[\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{2}-3 \sin ^{2} \theta \cos ^{2} \theta\right]-3 \cos ^{4} \theta$
$=1-3\left(1-\cos ^{2} \theta\right) \cos ^{2} \theta-3 \cos ^{4} \theta$
$=1-3+3 \cos ^{4} \theta-3 \cos ^{4} \theta$
$=-2$