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

Question

Aicharya · Accepted Answer

The $\sin ^6 \theta+\cos ^6 \theta-3 \cos ^4 \theta$ $=\left(\sin ^2 \theta+\cos ^2 \theta\right)\left(\sin ^4 \theta+\cos ^4 \theta-\sin ^2 \theta \cos ^2 \theta\right)-3 \cos ^4 \theta$ $\begin{aligned} & =1 \times\left[\left(\sin ^2 \theta+\cos ^2 \theta\right)^2-3 \sin ^2 \theta \cos ^2 \theta\right]-3 \cos ^4 \theta \ & =1-3\left(1-\cos ^2 \theta\right) \cos ^2 \theta-3 \cos ^4 \theta \ & =1-3+3 \cos ^4 \theta-3 \cos ^4 \theta=-2 \end{aligned}$ Hence, option (b) is correct.

The possible value of \(\sin ^6(\theta)+\cos ^6(\theta)-3 \cos ^4(\theta)\) is

Solution