Question

prove(1+cot theta -cosec theta)(1+sec theta+ tan theta)= 2

Answer 1

(1 + cos / sin - 1/sin ) ( 1 + 1/ cos + 1/sin )

(sin + cos -1  ) ( cos + sin + 1)
------------------------------
cos  * sin

(sin+cos)² - 1²
----------------------
cos  * sin

1 + 2 sin cos  - 1
-------------------------------
cos * sin

= 2

Answer 2

Given LHS
$(1 + cot\theta - cosec\theta)(1 + sec\theta + tan\theta)$

$( \frac{Sin\theta + cos\theta - 1}{sin\theta}) ( \frac{Cos\theta + 1 + Sin\theta}{sin\theta})$

$\frac{(sin\theta + cos\theta)^2 - 1^2}{sin\theta cos\theta}$

$\frac{sin^2\theta + cos^2\theta + 2sin\theta cos\theta - 1}{sin\theta cos\theta}$

$\frac{1 + 2sin\theta cos\theta - 1}{sin\theta cos\theta}$

= 2

= RHS