Trigonometric Identities Question 352
Question: If $ cosec\theta =\frac{p+q}{p-q}, $ then $ \cot ,( \frac{\pi }{4}+\frac{\theta }{2} )= $
[EAMCET 2001]
Options:
A) $ \sqrt{\frac{p}{q}} $
B) $ \sqrt{\frac{q}{p}} $
C) $ \sqrt{pq} $
D) $ pq $
Show Answer
Answer:
Correct Answer: B
Solution:
Given, $ cosec\theta =\frac{p+q}{p-q} $
Þ $ \frac{1}{\sin \theta }=\frac{p+q}{p-q} $ Apply componendo and dividendo $ \frac{1+\sin \theta }{1-\sin \theta }=\frac{p+q+p-q}{p+q-p+q} $
Þ $ {{{ \frac{\cos \frac{\theta }{2}+\sin \frac{\theta }{2}}{\cos \frac{\theta }{2}-\sin \frac{\theta }{2}} }}^{2}}=\frac{p}{q} $
Þ $ {{{ \frac{1+\tan \frac{\theta }{2}}{1-\tan \frac{\theta }{2}} }}^{2}}=\frac{p}{q} $
Þ $ {{\tan }^{2}}( \frac{\pi }{4}+\frac{\theta }{2} )=\frac{p}{q} $
Þ $ {{\cot }^{2}}( \frac{\pi }{4}+\frac{\theta }{2} )=\frac{q}{p} $ Note: $ \cot ( \frac{\pi }{4}+\frac{\theta }{2} )=\sqrt{\frac{q}{p}},\text{only,}if $ $ \cot ,( \frac{\pi }{4}+\frac{\theta }{2} )>0 $ .
BETA
