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Question: The value of \(\frac{\cot 54{^\circ}}{\tan 36{^\circ}} + \frac{\tan 20{^\circ}}{\cot 70{^\circ}}\) i...

The value of cot⁡54∘tan⁡36∘+tan⁡20∘cot⁡70∘\frac{\cot 54{^\circ}}{\tan 36{^\circ}} + \frac{\tan 20{^\circ}}{\cot 70{^\circ}} is

A

2

B

3

C

1

D

0

Answer

2

Explanation

Solution

tan⁡(90∘−θ)=cot⁡θ, 6mucot⁡(90∘−θ)=tan⁡θ.\tan(90{^\circ} - \theta) = \cot\theta,\mspace{6mu}\cot(90{^\circ} - \theta) = \tan\theta.

Therefore cot⁡54∘tan⁡36∘+tan⁡20∘cot⁡70∘\frac{\cot 54{^\circ}}{\tan 36{^\circ}} + \frac{\tan 20{^\circ}}{\cot 70{^\circ}} =cot⁡54∘tan⁡(90∘−54∘)+tan⁡20∘(cot⁡90∘−20∘)= \frac{\cot 54{^\circ}}{\tan(90{^\circ} - 54{^\circ})} + \frac{\tan 20{^\circ}}{(\cot 90{^\circ} - 20{^\circ})}

cot⁡54∘cot⁡54∘+tan⁡20∘tan⁡20∘=1+1=2\frac{\cot 54{^\circ}}{\cot 54{^\circ}} + \frac{\tan 20{^\circ}}{\tan 20{^\circ}} = 1 + 1 = 2.