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$\mathbf{a}$ and $\mathbf{b}$ are unit vectors such that $\mathbf{a}+2 \mathbf{b}$ is also a unit vector. If $\theta$ is the angle between a and $\mathbf{b}$, then $\sin \theta+\cos ^3 \theta+\tan ^5 \theta$ is equal to
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$\begin{aligned} & \text { Given, }|\mathbf{a}|=|\mathbf{b}|=|\mathbf{a}+2 \mathbf{b}|=1 \\ & \qquad \begin{aligned}|\mathbf{a}+2 \mathbf{b}|^2 & =|\mathbf{a}|^2+4\left|\mathbf{b}^2\right|+4|\mathbf{a}||\mathbf{b}| \cos \theta \\ 1 & =1+4+4 \cos \theta \\ \cos \theta & =-1 \Rightarrow \theta=\pi \\ \sin \theta & +\cos ^3 \theta+\tan ^3 \theta \\ & =\sin \pi+\cos ^3 \pi+\tan ^3 \pi \\ =0 & +(-1)^3+0=-1\end{aligned}\end{aligned}$
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