If $\operatorname{cosec} x=1+\cot x$, then $x=2 n \pi, 2 n \pi+\frac{\pi}{2}$
If $\tan \theta+\tan 2 \theta+\sqrt{3} \tan \theta \tan 2 \theta=\sqrt{3}$, then $\theta=\frac{n \pi}{3}+\frac{\pi}{9}$.
If $\tan (\pi \cos \theta)=\cot (\pi \sin \theta)$, then $\cos \left(\theta-\frac{\pi}{4}\right)= \pm \frac{1}{2 \sqrt{2}}$.
In the following match each item given under the Column I to its correct answer given under the Column II.
| Column I | Column II | ||
|---|---|---|---|
| (i) | $\sin (x+y) \sin (x-y)$ |
(a) | $\cos ^2 x-\sin ^2 y$ |
| (ii) | $\cos (x+y) \cos (x-y)$ | (b) | $1-\tan \theta / 1+\tan \theta$ |
| (iii) | $\cot \left(\frac{\pi}{4}+\theta\right)$ | (c) | $1+\tan \theta / 1-\tan \theta$ |
| (iv) | $\tan \left(\frac{\pi}{4}+\theta\right)$ | (d) | $\sin ^2 x-\sin ^2 y$ |
(i) $\sin (x+y) \sin (x-y)=\sin ^2 x-\sin ^2 y$
(ii) $\cos (x+y) \cos (x-y)=\cos ^2 x-\sin ^2 y$
(iii) $$\begin{aligned} \cot \left(\frac{\pi}{4}+\theta\right) & =\frac{\cot \frac{\pi}{4} \cot \theta-1}{\cot \frac{\pi}{4}+\cot \theta} \\ & =\frac{-1+\cot \theta}{1+\cot \theta}=\frac{1-\tan \theta}{1+\tan \theta} \end{aligned}$$
(iv) $\tan \left(\frac{\pi}{4}+\theta\right)=\frac{\tan \frac{\pi}{4}+\tan \theta}{1-\tan \frac{\pi}{4} \tan \theta}=\frac{1+\tan \theta}{1-\tan \theta}$
Hence, the correct mathes are (i) $\rightarrow$ (d), (ii) $\rightarrow$ (a), (iii) $\rightarrow$ (b), (iv) $\rightarrow$ (c).