Question

prove that (tan theta +cot theta +1 ) (tan theta + cot theta‐1) = sec theta cot theta
pls explain too hard​

Answer 1

It should have been sec²θ+cot²θ instead of secθ + cosθ. It would be incorrect if it was the latter. Considering that you intended to ask (tanθ + cotθ + 1) (tanθ + cotθ - 1) = sec²θ+cot²θ, here's your answer.

Proving:

=> (tanθ + cotθ + 1) (tanθ + cotθ - 1)

We will use the property (a+b)(a-b) = a²-b². Here, tanθ + cotθ is a and 1 is b.

=> (tanθ + cotθ)² - (1)²

Using the property (a+b)² = a² + b² + 2ab, we get

=> tan²θ + cot²θ + 2tanθcotθ - 1

=> (sec²θ - 1) + (cot²θ) + 2tanθ x 1/tanθ - 1

=> sec²θ -1 + cot²θ + 2 - 1

=> sec²θ + cot²θ

Hence proved