Question

If $\cos ec\theta =(p+q)/(p-q)$ , then $cot\left[\right(\pi /4)+(\theta /2\left)\right]=?$

• A. $\sqrt{(p/q)}$
• B. $\sqrt{(q/p)}$
• C. $\sqrt{pq}$
• D. $pq$

Answer 1

The correct option is B

$\sqrt{(q/p)}$

Explanation for the correct option:

Step 1. Expressing the given data:

Given data, $\cos ec\theta =(p+q)/(p-q)$

Let this is the first equation:

$\cos ec\theta =(p+q)/(p–q)\dots .\left(i\right)$

So we can write it like

$1/\sin \theta =(p+q)/(p–q)$

Step 2. Using componendo and dividendo rule:

⇒ $(1+\sin \theta )/(1–\sin \theta )=(p+q+p–q)/(p+q–p+q)$

⇒ ${\left[\right(\cos \theta /2+\sin \theta /2)/(\cos \theta /2–\sin \theta /2\left)\right]}^2=2p/2q$

⇒ $\left[\right(1+\tan \theta /2)/(1–\tan \theta /2\left)\right]2=p/q$

⇒$\left[\right(\tan \pi /4+\tan \theta /2)/(1–\tan \pi /4\tan \theta /2\left)\right]2=p/q$

⇒ ${\tan }^2(\pi /4+\theta /2)=p/q$

Step 3. We will take under root of both side

$\tan (\pi /4+\theta /2)=\sqrt{(p/q)}$

So , we got the $\mathit{c}\mathit{o}\mathit{t}\mathbf{(}\mathit{\pi }\mathbf{/}\mathbf{4}\mathbf{}\mathbf{+}\mathbf{}\mathit{\theta }\mathbf{/}\mathbf{2}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\sqrt{\mathbf{(}\mathbf{q}\mathbf{/}\mathbf{p}\mathbf{)}}$

Therefore , Option (B) is correct.