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If sin θ + cos θ = √3, then the value of tan θ + cot θ =

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answer is A.

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Detailed Solution

Given that,

sin θ + cos θ = 3 … …

. (1)
Apply square on both sides of eq (1) and use

(a + b) 2 = a 2 + b 2 + 2 a b

(sin θ + cos θ) 2 = (3) 2 sin 2 θ + cos 2 θ + 2 sin θ ⋅ cos θ = 3 … … .. (2)

Now, substitute, sin² θ + cos² θ = 1 in eq (2)

1 + 2 sin θ, cos θ = 3 2 sin θ ⋅ cos θ = 3 − 1 2 sin θ ⋅ cos θ = 2 sin θ . cos θ = 22 sin θ ⋅ cos θ = 1 … … .. (3)

We need to find the value of tan θ + cot θ.
Substitute

tan θ = sin θ cos θ

and

cot θ = cos θ sin θ

in the expression tan θ + cot θ

= tan θ + cot θ = sin θ cos θ + cos θ sin θ = sin 2 θ + cos 2 θ sin θ ⋅ cos θ … … .. 4

Substitute identity sin² θ + cos² θ = 1 and the value from eq (3). We get,

sin 2 θ + cos 2 θ sin θ ⋅ cos θ = 11 = 1

So, tan θ + cot θ = 1
The correct option is (1).

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