| The value of\[\frac{\tan 27{}^\circ +\cot 63{}^\circ }{\tan 27{}^\circ \,\,(\sin 25{}^\circ +\cos 65{}^\circ )}\]is [SSC (CPO) 2013] |
A) \[\text{cosec}\,\,25{}^\circ \]
B) \[2\,\,\tan 27{}^\circ \]
C) \[\sin 25{}^\circ \]
D) \[\tan 65{}^\circ \]
Correct Answer: A
Solution :
| Given, \[\frac{\tan 27{}^\circ +\cot 63{}^\circ }{\tan 27{}^\circ \,\,(\sin 25{}^\circ +\cos 65{}^\circ )}\] |
| \[=\frac{\tan 27{}^\circ +\cot \,\,(90{}^\circ -27{}^\circ )}{\tan 27{}^\circ \,\,[\sin 25{}^\circ +\cos \,\,(90{}^\circ -25{}^\circ )]}\] |
| \[[\because \cot \,\,(90{}^\circ -\theta )=tan\theta \,\,and\,\,cos\,\,(90{}^\circ -\theta )=\sin \theta ]\] |
| \[=\frac{\tan 27{}^\circ +\tan 27{}^\circ }{\tan 27{}^\circ \,\,(\sin 25{}^\circ +\sin 25{}^\circ )}\] |
| \[=\frac{2\tan 27{}^\circ }{\tan 27{}^\circ \,\,(2\sin 25{}^\circ )}=\text{cosec}25{}^\circ \] |
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