Question

If $\alpha ,\beta$ and $\gamma$ belongs to $[0,\pi ]$ and $\alpha ,\beta ,\gamma$ are in AP, then $\frac{\sin \alpha -\sin \gamma }{\cos \gamma -\cos \alpha }$ is equal to

• A. $\sin \beta$
• B. $\cos \beta$
• C. $cot\beta$
• D. $\cos ec\beta$
• E. $2\cos \beta$

Answer 1

The correct option is C

$cot\beta$

Explanation for the correct option:

Step 1. Find the value of $\frac{\sin \alpha -\sin \gamma }{\cos \gamma -\cos \alpha }$:

Given that $\alpha ,\beta ,\gamma$ are in AP, then

$\Rightarrow$$2\beta =\alpha +\gamma$

$\Rightarrow$ $\beta =\frac{(\alpha +\gamma )}{2}$

Step 2. By using the identities $\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{x}\mathbf{}\mathbf{–}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{y}\mathbf{}\mathbf{=}\mathbf{}\mathbf{2}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\frac{\mathbf{(}\mathbf{x}\mathbf{}\mathbf{+}\mathbf{}\mathbf{y}\mathbf{)}}{\mathbf{2}}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\frac{\mathbf{(}\mathbf{x}\mathbf{}\mathbf{–}\mathbf{}\mathbf{y}\mathbf{)}}{\mathbf{2}}\mathbf{,}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{x}\mathbf{}\mathbf{–}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{y}\mathbf{}\mathbf{=}\mathbf{}\mathbf{-}\mathbf{2}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\frac{\mathbf{(}\mathbf{x}\mathbf{}\mathbf{+}\mathbf{}\mathbf{y}\mathbf{)}}{\mathbf{2}}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\frac{\mathbf{(}\mathbf{x}\mathbf{}\mathbf{–}\mathbf{}\mathbf{y}\mathbf{)}}{\mathbf{2}}$, we get

$\begin{array}{rcl}\frac{\sin \alpha –\sin \gamma }{\cos \gamma –\cos \alpha }& =& \frac{2\cos {\displaystyle \frac{(\alpha +\gamma )}{2}}\sin {\displaystyle \frac{(\alpha –\gamma )}{2}}}{-2\sin \frac{(\gamma +\alpha )}{2}\sin \frac{(\gamma –\alpha )}{2}}\\ & =& \frac{2\cos {\displaystyle \frac{(\alpha +\gamma )}{2}}\sin {\displaystyle \frac{(\alpha –\gamma )}{2}}}{2\sin \frac{(\gamma +\alpha )}{2}\sin \frac{(\alpha –\gamma )}{2}}\\ & =& \frac{\cos {\displaystyle \frac{(\alpha +\gamma )}{2}}}{\sin \frac{(\alpha +\gamma )}{2}}\\ & =& cot\frac{(\alpha +\gamma )}{2}\\ & =& \mathit{c}\mathit{o}\mathit{t}\mathbf{}\mathit{\beta }\end{array}$

Hence, Option ‘C’ is Correct.