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JEE Main
Exam
JEE Main
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

JEE Main · Mathematics · Trigonometric Ratios & Identities

81^ ^ ^ - 63^ ^ ^ - 27^ ^ ^ + 9^ ^ ^ equals to

tan⁡81∘−tan⁡63∘−tan⁡27∘+tan⁡9∘\tan 81^{{}^{{}^{\circ}}}-\tan 63^{{}^{{}^{\circ}}}-\tan 27^{{}^{{}^{\circ}}}+\tan 9^{{}^{{}^{\circ}}} equals to

Options

  1. A66
  2. B00
  3. C22
  4. D44

Correct answer

D. 44

Step-by-step solution

tan⁡81∘−tan⁡63∘−tan⁡27∘+tan⁡9∘=(tan⁡81∘+tan⁡9∘)−[tan⁡(63∘)+tan⁡(27∘)]=[sin⁡81∘cos⁡81∘+sin⁡9∘cos⁡9∘]−[sin⁡63∘cos⁡63∘+sin⁡27∘cos⁡27∘]=[sin⁡81∘cos⁡9∘+sin⁡9∘cos⁡81∘cos⁡9∘cos⁡81∘]−[sin⁡63∘cos⁡27∘+sin⁡27∘cos⁡63∘cos⁡27∘cos⁡63∘]=[sin⁡(81∘+9∘)]cos⁡9∘cos⁡(90∘−90∘)=[sin⁡(63∘+27∘)]cos⁡27∘cos⁡(90∘−27∘)=1sin⁡9∘cos⁡9∘×22−1sin⁡27∘cos⁡27∘×22=2cos⁡27∘sin⁡27∘−2sin⁡54∘\begin{aligned} & \tan 81^{{}^{{}^{\circ}}}-\tan 63^{{}^{{}^{\circ}}}-\tan 27^{{}^{{}^{\circ}}}+\tan 9^{{}^{{}^{\circ}}} \\ & =\left(\tan 81^{{}^{{}^{\circ}}}+\tan 9^{{}^{{}^{\circ}}}\right)-\left[\tan \left(63^{{}^{{}^{\circ}}}\right)+\tan \left(27^{{}^{{}^{\circ}}}\right)\right] \\ & =\left[\frac{\sin 81^{{}^{{}^{\circ}}}}{\cos 81^{{}^{{}^{\circ}}}}+\frac{\sin 9^{{}^{{}^{\circ}}}}{\cos 9^{{}^{{}^{\circ}}}}\right]-\left[\frac{\sin 63^{{}^{{}^{\circ}}}}{\cos 63^{{}^{{}^{\circ}}}}+\frac{\sin 27^{{}^{{}^{\circ}}}}{\cos 27^{{}^{{}^{\circ}}}}\right] \\ & =\left[\frac{\sin 81^{{}^{{}^{\circ}}} \cos 9^{{}^{{}^{\circ}}}+\sin 9^{{}^{{}^{\circ}}} \cos 81^{{}^{{}^{\circ}}}}{\cos 9^{{}^{{}^{\circ}}} \cos 81^{{}^{{}^{\circ}}}}\right] \\ & -\left[\frac{\sin 63^{{}^{{}^{\circ}}} \cos 27^{{}^{{}^{\circ}}}+\sin 27^{{}^{{}^{\circ}}} \cos 63^{{}^{{}^{\circ}}}}{\cos 27^{{}^{{}^{\circ}}} \cos 63^{{}^{{}^{\circ}}}}\right] \\ & =\frac{\left[\sin \left(81^{{}^{{}^{\circ}}}+9^{{}^{{}^{\circ}}}\right)\right]}{\cos 9^{{}^{{}^{\circ}}} \cos \left(90^{{}^{{}^{\circ}}}-90^{{}^{{}^{\circ}}}\right)} \\ & =\frac{\left[\sin \left(63^{{}^{{}^{\circ}}}+27^{{}^{{}^{\circ}}}\right)\right]}{\cos 27^{{}^{{}^{\circ}}} \cos \left(90^{{}^{{}^{\circ}}}-27^{{}^{{}^{\circ}}}\right)} \\ & =\frac{1}{\sin 9^{{}^{{}^{\circ}}} \cos 9^{{}^{{}^{\circ}}}} \times \frac{2}{2}-\frac{1}{\sin 27^{{}^{{}^{\circ}}} \cos 27^{{}^{{}^{\circ}}}} \times \frac{2}{2} \\ & =\frac{2}{\cos 27^{{}^{{}^{\circ}}} \sin 27^{{}^{{}^{\circ}}}}-\frac{2}{\sin 54^{{}^{{}^{\circ}}}}\end{aligned}=2[sin⁡54∘−sin⁡18∘sin⁡18∘sin⁡54∘]=2[2cos⁡36∘sin⁡18∘sin⁡18∘cos⁡36∘]=4\begin{aligned} & =2\left[\frac{\sin 54^{{}^{{}^{\circ}}}-\sin 18^{{}^{{}^{\circ}}}}{\sin 18^{{}^{{}^{\circ}}} \sin 54^{{}^{{}^{\circ}}}}\right] \\ & =2\left[\frac{2 \cos 36^{{}^{{}^{\circ}}} \sin 18^{{}^{{}^{\circ}}}}{\sin 18^{{}^{{}^{\circ}}} \cos 36^{{}^{{}^{\circ}}}}\right] \\ & =4\end{aligned}$
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