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JEE Advanced
Exam
JEE Advanced
Subject
Mathematics
Chapter
Trigonometry

JEE Advanced · Mathematics · Trigonometry

Let L be the least value of the expression ( 2 x- 2 x)/(1+ ((5...

Let LL be the least value of the expression cot⁡2x−tan⁡2x1+sin⁡(5π2−8x)\frac{\cot 2 x-\tan 2 x}{1+\sin \left(\frac{5 \pi}{2}-8 x\right)} if x∈(0,π8)x \in\left(0, \frac{\pi}{8}\right) and MM denotes the value of 4(cos⁡310∘+sin⁡320∘)(cos⁡10∘+sin⁡20∘)\frac{4\left(\cos ^3 10^{\circ}+\sin ^3 20^{\circ}\right)}{\left(\cos 10^{\circ}+\sin 20^{\circ}\right)}. Find the value of (L+M)(L+M)

Correct answer

05

Step-by-step solution

L: Expression reduces to 2cosec⁡8x2 \operatorname{cosec} 8 x ⇒L=2\Rightarrow \quad \mathrm{L}=\mathbf{2} M: We have 4(cos⁡310∘+sin⁡320∘)(cos⁡10∘+sin⁡20∘)=4cos⁡310∘+4sin⁡320∘cos⁡10∘+sin⁡20∘=cos⁡30∘+3cos⁡10∘+3sin⁡20∘−sin⁡60∘cos⁡10∘+sin⁡20∘=3(cos⁡10∘+sin⁡20∘)(cos⁡10∘+sin⁡20∘)=3=M Ans. \begin{aligned} & \frac{4\left(\cos ^3 10^{\circ}+\sin ^3 20^{\circ}\right)}{\left(\cos 10^{\circ}+\sin 20^{\circ}\right)}=\frac{4 \cos ^3 10^{\circ}+4 \sin ^3 20^{\circ}}{\cos 10^{\circ}+\sin 20^{\circ}}=\frac{\cos 30^{\circ}+3 \cos 10^{\circ}+3 \sin 20^{\circ}-\sin 60^{\circ}}{\cos 10^{\circ}+\sin 20^{\circ}} \\ & =\frac{3\left(\cos 10^{\circ}+\sin 20^{\circ}\right)}{\left(\cos 10^{\circ}+\sin 20^{\circ}\right)}=3=\mathrm{M} \text { Ans. }\end{aligned} Hence L+M=2+3=5L+M=2+3=5
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