Question

Prove (1+sin/csc-cot)-(1-sin/csc+cot)=2(1+cot)​

Answer 1

$\dfrac{1+sin}{csc-cot}-\dfrac{1-sin}{csc+cot}=2(1+cot)\\\\\\\text{Proof from LHS }\rightarrow \text{ RHS}\\\\\dfrac{1+sin}{csc-cot}\bigg(\dfrac{csc+cot}{csc+cot}\bigg)-\dfrac{1-sin}{csc+cot}\bigg(\dfrac{csc-cot}{csc-cot}\bigg)\\\\\\\dfrac{csc+cot+sin csc+sincot}{csc^2-cot^2}+\dfrac{-csc+cot+sincsc-sincot}{csc^2-cot^2}\\\\\\\dfrac{2cot+2sincsc}{csc^2-cot^2}\\\\\\\dfrac{2(cot+sincsc)}{1}\\\\\\2\bigg(\dfrac{cos}{sin}+\dfrac{sin\cdot1}{sin}\bigg)\\\\\\2(cot+1)\\\\\\2(1+cot)=2(1+cot)$

LHS = RHS $\checkmark$