If sin θ + cos θ = √2, then evaluate : tan θ + cot θ

Given: sin θ + cos θ = √2 …(i)

We need to evaluate tan θ + cot θ

We know tan θ and cot θ can be expressed in sin θ and cos θ as,

tan θ =

& cot θ =

So, tan θ + cot θ =

=

=

We have got an equation, tan θ + cot θ = …(ii)

Squaring equation (i) on both sides,

(sin θ + cos θ)² = (√2)²

⇒ sin² θ + cos² θ + 2 sin θ cos θ = 2

⇒ 1 + 2 sin θ cos θ = 2 [∵ sin² θ + cos² θ = 1]

⇒ 2 sin θ cos θ = 2 – 1

⇒ sin θ cos θ = 1/2 …(iii)

Putting the value of equation (iii) in equation (ii), we get

tan θ + cot θ =

⇒ tan θ + cot θ = 2

Hence, tan θ + cot θ = 2.