Question

If $2sec2\alpha =\tan \beta +cot\beta$, then one of the value of $\alpha +\beta$ is

• A. $\frac{\mathrm{\pi }}{4}$
• B. $\frac{\mathrm{\pi }}{2}$
• C. $\mathrm{\pi }$
• D. $n\pi –\frac{\pi }{4},n\in I$

Answer 1

The correct option is A

$\frac{\mathrm{\pi }}{4}$

Explanation for the correct option:

Find the value of $\alpha +\beta$:

Given that,

$2sec2\alpha =\tan \beta +cot\beta$

$\Rightarrow$ $2sec2\alpha =\frac{\sin \beta }{\cos \beta }+\frac{\cos \beta }{\sin \beta }$

$\Rightarrow$ $2sec2\alpha =\frac{(\sin ²\beta +\cos ²\beta )}{\sin \beta \cos \beta }$

$\Rightarrow$ $2sec2\alpha =\frac{1}{\sin \beta \cos \beta }$ $\left[\mathbf{\because }\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{\theta }\mathbf{}\mathbf{+}\mathbf{}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{\theta }\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\right]$

$\Rightarrow$ $sec2\alpha =\frac{1}{2\sin \beta \cos \beta }$

$\Rightarrow$ $sec2\alpha =\frac{1}{\sin 2\beta }$ $\left[\mathbf{\because }\mathbf{2}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\theta }\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{\theta }\mathbf{}\mathbf{=}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathbf{2}\mathit{\theta }\right]$

$\Rightarrow$ $\sin 2\beta =\frac{1}{sec2\alpha }$

$\Rightarrow$ $\sin 2\beta =\cos 2\alpha$

$\Rightarrow$ $2\beta =\left(\frac{\pi }{2}–2\alpha \right)$ $\left[\mathbf{\because }\mathit{c}\mathit{o}\mathit{s}\left(\frac{\pi }{2}-\theta \right)\mathbf{}\mathbf{=}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\theta }\right]$

$\Rightarrow$$2\beta +2\alpha =\frac{\pi }{2}$

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

$\Rightarrow$ $\mathit{\alpha }\mathbf{}\mathbf{+}\mathbf{}\mathit{\beta }\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{\pi }}{\mathbf{4}}$

Hence, Option ‘A’ is Correct.