Question

12. If tan0 + cot0 = 5, find the value of tan²0 + cot?0.​

Answer 1

Given :

tanθ + cotθ = 5

To find :

tan²θ + cot²θ

Identity used :

• (a+b)² = a² + b² + 2ab
• tanθ = sinθ ÷ cosθ
• cotθ = cosθ ÷ sinθ

Solution :

We are given that, tanθ + cotθ = 5

On squaring both side, we get;

$\sf \implies \: ( \: \tan ( \theta) \: + \: \cot( \theta) )^{2} \: = \: {5}^{2} \\ \\ \sf \implies \: \: \tan^{2} ( \theta) + \: \cot^{2} ( \theta) + 2 \times \tan ( \theta) \times \cot( \theta) \: = \: 25 \\ \\ \sf \implies \: \: \tan^{2} ( \theta) \: + \: \cot^{2} ( \theta) + \: \: 2 \times \dfrac{ \sin( \theta) }{ \cos( \theta) } \times \dfrac{ \cos( \theta) }{ \sin( \theta) } \: = 25 \\ \\ \sf \implies \: \: \tan^{2} ( \theta) \: + \: \cot^{2} ( \theta) + \: \: 2 \times 1 \times 1 \: = \: 25 \\ \\ \sf \implies \: \: \tan^{2} ( \theta) \: + \: \cot^{2} ( \theta) + \: \: 2 \: = \: 25 \\ \\ \sf \implies \: \: \tan^{2} ( \theta) \: + \: \cot^{2} ( \theta) \: = \: 25 - 2 \\ \\ \sf \implies \: \: \tan^{2} ( \theta) \: + \: \cot^{2} ( \theta) \: = \: 23$

ANSWER :

tan²θ + cot²θ = 23