If cot θ + tan θ = 2 cosec θ,

Question:

If cot θ + tan θ = 2 cosec θ, then find the general value of θ.

Solution:

According to the question,

$\Rightarrow \frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{\cos \theta}=2 \operatorname{cosec} \theta$

Since,

$\sin ^{2} \theta+\cos ^{2} \theta=1$

$\Rightarrow \frac{\cos ^{2} \theta+\sin ^{2} \theta}{\sin \theta \cos \theta}=2 \operatorname{cosec} \theta$

⇒ 1 = 2 cosec θ sin θ cos θ

We know that,

sin θ cosec θ = 1

⇒ 1 = 2 cos θ

⇒ cos θ = 1/2 = cos(π/3)

Hence,

The solution of cos x = cos α can be given by,

x = 2mπ ± α ∀ m ∈ Z

⇒ θ = 2nπ ± π/3, n ∈ Z

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