Question

The value of the expression $1+cosec\frac{\pi }{4}+cosec\frac{\pi }{8}+cosec\frac{\pi }{16}$ is equal to

• A. $cot\frac{\pi }{8}$
• B. $cot\frac{\pi }{16}$
• C. $cot\frac{\pi }{32}$
• D. $c o s e c^{2}\frac{\pi }{16}$

Answer 1

Answer: (C)

We know that
$cosec\theta +cot \theta =\frac{1 + cos ⁡ \theta }{sin ⁡ \theta }=\frac{2 c o s^{2} \frac{\theta }{2}}{2 s i n \frac{\theta }{2} c o s \frac{\theta }{2}}=cot\frac{\theta }{2}$
$\Rightarrow cosec\theta =cot\frac{\theta }{2}-cot \theta $
$\Rightarrow cosec\frac{\pi }{4}=cot\frac{\pi }{8}-cot\frac{\pi }{4}$
$cosec\frac{\pi }{8}=cot\frac{\pi }{16}-cot\frac{\pi }{8}$
$cosec\frac{\pi }{16}=cot\frac{\pi }{32}-cot\frac{\pi }{16}$
Adding them, we get,
$cosec\frac{\pi }{4}+cosec\frac{\pi }{8}+cosec\frac{\pi }{16}=cot\frac{\pi }{32}-cot\frac{\pi }{4}$
$\Rightarrow 1+cosec\frac{\pi }{4}+cosec\frac{\pi }{8}+cosec\frac{\pi }{16}=cot\frac{\pi }{32}$