Question

If tan θ + cot θ = 2, prove that tan²θ + cot²θ = 2.

Answer 1

Given: tan θ + cot θ = 2

To Prove: tan²θ + cot²θ = 2

Proof [Step-wise]:

1. Put t = tan θ.

The given becomes `t + 1/t = 2`.

2. Multiply by t:

t² + 1 = 2t 

⇒ t² – 2t + 1 = 0

⇒ (t – 1)² = 0 

⇒ t = 1

3. So, tan θ = 1

⇒ `θ = π/4 + kπ (k ∈ Z)` 

Then `2θ = π/2 + 2kπ`, for which tan (2θ) is undefined (blows up) and cot (2θ) = 0. 

Hence, tan²θ + cot²θ is not a finite value and therefore cannot equal 2.

So, the statement tan²θ + cot²θ = 2 is false as written.