If tanθ + (1/ tanθ) =2 then show that tan²θ + (1 / tan²θ)= 2

Solution

We are given that:

tanθ + (1/ tanθ) = 2

Squaring both sides, we get:

(tanθ + (1/ tanθ))^2 = 2^2

Expanding the left-hand side using the identity (a + b)^2 = a^2 + 2ab + b^2, we get:

tan^2θ + 2(1)(tanθ)(1/tanθ) + 1/tan^2θ = 4

Simplifying, we get:

tan^2θ + 1/tan^2θ + 2 = 4

tan^2θ + 1/tan^2θ = 2

Therefore, we have shown that tan²θ + (1/tan²θ) = 2, given that tanθ + (1/tanθ) = 2.