Question

Evaluate $:\frac{\cot 54°}{\tan 36°}+\frac{\tan 20°}{\cot 70°}-2\tan 45°$

Answer 1

Find the value of $\frac{\cot 54°}{\tan 36°}+\frac{\tan 20°}{\cot 70°}-2\tan 45°$

By using trigonometric ratios

$\tan \left(90°-x\right)=\cot x$

$\cot \left(90°-x\right)=\tan x$

Equation becomes

$\frac{\cot \left(90°-54\right)}{\tan 36°}+\frac{\tan 20°}{\cot \left(90°-70\right)}-2\tan 45°$

$\Rightarrow \frac{\tan {36}^{\circ }}{\tan 36°}+\frac{\tan 20°}{\tan {20}^{\circ }}-2\tan 45°$

$\Rightarrow 1+1-2\tan {45}^{\circ }$

$\Rightarrow 2-2\tan {45}^{\circ }$ $(\tan {45}^{\circ }=1)$

So,

$\Rightarrow 2-2\times 1=0$

Hence, the value of $\frac{\cot 54°}{\tan 36°}+\frac{\tan 20°}{\cot 70°}-2\tan 45°=0$