Question

If $\cos x=3\cos y$, then $2\tan \left[\frac{(y-x)}{2}\right]$ is equal to

• A. $cot\left[\frac{(y–x)}{2}\right]$
• B. $cot\left[\frac{(x+y)}{4}\right]$
• C. $cot\left[\frac{(y–x)}{4}\right]$
• D. $cot\left[\frac{(x+y)}{2}\right]$

Answer 1

The correct option is D

$cot\left[\frac{(x+y)}{2}\right]$

Explanation for the correct option:

Step 1. Find the value of $2\tan \left[\frac{(y-x)}{2}\right]$:

Given, $\cos x=3\cos y$

$\Rightarrow$ $\frac{\cos x}{\cos y}=\frac{3}{1}$

Step 2. By Using componendo and dividendo rule, we get

$\frac{(\cos x+\cos y)}{(\cos x–\cos y)}=\frac{(3+1)}{(3–1)}$

$\Rightarrow$$\frac{2\cos {\displaystyle \frac{(x+y)}{2}}\cos {\displaystyle \frac{(x–y)}{2}}}{2\sin \frac{(x+y)}{2}\sin \frac{(y–x)}{2}}=\frac{4}{2}$

$\Rightarrow$ $\left[cot\frac{(x+y)}{2}\right]\left[cot\frac{(y–x)}{2}\right]=2$

$\Rightarrow$ $cot\frac{(x+y)}{2}=\frac{2}{\left[cot{\displaystyle \frac{(y–x)}{2}}\right]}$

$\mathbf{\therefore }\mathbf{2}\mathbf{}\mathit{t}\mathit{a}\mathit{n}\frac{\mathbf{(}\mathbf{y}\mathbf{}\mathbf{–}\mathbf{}\mathbf{x}\mathbf{)}}{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}\mathit{c}\mathit{o}\mathit{t}\frac{\mathbf{(}\mathbf{x}\mathbf{}\mathbf{+}\mathbf{}\mathbf{y}\mathbf{)}}{\mathbf{2}}$

Hence, Option ‘D’ is Correct.