Question

If `sin θ + cos θ = sqrt(2)`, prove that tan θ + cot θ = 2.

Answer 1

Given, `sin θ + cos θ = sqrt(2)`

On squaring both the sides, we get

`(sin θ + cos θ)^2 = (sqrt(2))^2`

⇒ sin² θ + cos² θ + 2 sin θ cos θ = 2

⇒ 1 + 2 sin θ cos θ = 2

⇒ 2 sin θ cos θ = 1

⇒ `sin θ cos θ = 1/2`   ...(i)

Now taking L.H.S.,

`tan θ + cot θ = sin θ/cos θ + cos θ/sin θ`

= `(sin^2θ + cos^2θ)/(cosθ sinθ)`

= `1/(sin θ cos θ)`

= `1/(1//2)`   ...[From equation (i)]

= 2 = R.H.S.