A) \[\tan \left( \frac{\theta }{2}-\frac{\pi }{4} \right)\]
B) \[\tan \left( -\frac{\pi }{4}-\frac{\theta }{2} \right)\]
C) \[\tan \left( \frac{\pi }{4}-\frac{\theta }{2} \right)\]
D) \[\tan \left( \frac{\pi }{4}+\frac{\theta }{2} \right)\]
Correct Answer: C
Solution :
\[\frac{\cos \theta }{1+\sin \theta }=\frac{\sin \left( \frac{\pi }{2}-\theta \right)}{1+\cos \left( \frac{\pi }{2}-\theta \right)}\] \[=\frac{2\sin \left( \frac{\pi }{4}-\frac{\theta }{2} \right)\cos \left( \frac{\pi }{4}-\frac{\theta }{2} \right)}{2{{\cos }^{2}}\left( \frac{\pi }{4}-\frac{\theta }{2} \right)}\] \[=\frac{\sin \left( \frac{\pi }{4}-\frac{\theta }{2} \right)}{\cos \left( \frac{\pi }{2}-\frac{\theta }{2} \right)}\] \[=\tan \left( \frac{\pi }{4}-\frac{\theta }{2} \right)\] Alternate \[\frac{\cos \theta }{1+\sin \theta }\] \[=\frac{({{\cos }^{2}}\theta \text{/}2-{{\sin }^{2}}\theta \text{/}2)}{({{\sin }^{2}}\theta \text{/}2+{{\cos }^{2}}\theta \text{/}2)+2\sin \theta \text{/}2+\cos \theta \text{/}2}\] \[=\frac{(\cos \theta \text{/}2+\sin \theta \text{/}2)\,(\cos \theta \text{/}2-\sin \theta \text{/}2)}{{{(\cos \theta \text{/}2+\sin \theta \text{/}2)}^{2}}}\] \[=\frac{1-\tan \theta \text{/}2}{1+\tan \theta \text{/}2}\] \[=\frac{\tan \pi \text{/}4-tan\theta \text{/}2}{1+\tan \pi \text{/}4\cdot \tan \theta \text{/}2}\] \[=\tan \left( \frac{\pi }{4}-\frac{\theta }{2} \right)\]
You need to login to perform this action.
You will be redirected in
3 sec
