Question
In the following match each item given under the column C1 to its correct answer given under the column C2:
Column C1 Column C2
(a) $\sin(\text{x + y})\sin\text{x}-\text{y}$ (i) $\cos^2\text{x}-\sin^2\text{y}$
(b) $\cos(\text{x + y})\cos(\text{x}-\text{y})$ (ii) $\frac{1-\tan\theta}{1+\tan\theta}$
(c) $\cot\Big(\frac{\pi}{4}+\theta\Big)$ (iii) $\frac{1+\tan\theta}{1-\tan\theta}$
(d) $\tan\Big(\frac{\pi}{4}+\theta\Big)$ (iv) $\sin^2\text{x}-\sin^2\text{y}$

Answer

Column C1 Column C2
(a) $\sin(\text{x + y})\sin\text{x}-\text{y}$ (iv) $\sin^2\text{x}-\sin^2\text{y}$
(b) $\cos(\text{x + y})\cos(\text{x}-\text{y})$ (i) $\cos^2\text{x}-\sin^2\text{y}$
(c) $\cot\Big(\frac{\pi}{4}+\theta\Big)$ (ii) $\frac{1-\tan\theta}{1+\tan\theta}$
(d) $\tan\Big(\frac{\pi}{4}+\theta\Big)$ (ii) $\frac{1+\tan\theta}{1-\tan\theta}$
Explanation:
  1. $\sin(\text{x + y})\sin(\text{x}-\text{y})=\sin^2\text{x}-\sin^2\text{y}$
  2. $\cos(\text{x + y})\cos(\text{x}-\text{y})\cos^2\text{x}-\cos^2\text{y}$
  3. $\cot\Big(\frac{\pi}{4}+\theta\Big)=\frac{\frac{\cot\pi}{4}\cot\theta-1}{\cot\theta+\cot\frac{\pi}{4}}$
$=\frac{\cot\theta-1}{\cot\theta+1}=\frac{1-\tan\theta}{1+\tan\theta}$
  1. $\tan\Big(\frac{\pi}{4}+\theta\Big)=\frac{\tan\frac{\pi}{4}+\tan\theta}{1-\tan\frac{\pi}{4}\tan\theta}=\frac{1+\tan\theta}{1-\tan\theta}$

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