$\sin ^{2} 1+\sin ^{2} 3+\sin ^{2} 87+\sin ^{2} 89=\ldots \ldots \ldots \ldots$

If $\sin \theta+\cos \theta=p$ and $\sec \theta+\operatorname{cosec} \theta=q,$ then prove that $q\left(p^{2}-1\right)=2 p$

Write 'True' or 'False' and justify your answer.

$\cos \theta=\frac{a^{2} b^{2}}{2 a b},$ where $a$ and $b$ are two distinct numbers such that $a b>0$

The value of $\tan 5 \cdot \tan 25 \cdot \tan 45 \cdot \tan 65 \cdot \tan 85$ is $\ldots \ldots \ldots \ldots .$.

Write 'True' or 'False' and justify your answer.

$\sqrt{\left(1-\cos ^{2} \theta\right) \sec ^{2} \theta}=\tan \theta$