Question

Prove:

$\frac{cotA+cosecA-1}{cotA-cosecA+1}=\frac{1+cosA}{sinA}=cosec\theta +cot\theta =\frac{sinA}{1-cosA}$

Answer 1

$\frac{cosecA+cotA-1}{cotA-cosecA+1}$

we know that,$cosec²A-cot²A=1$

substituting this in the numerator,

$\frac{cosecA+cotA-(cosec²A-cot²A)}{(cotA-cosecA+1)}$

$x²-y²=(x+y)(x-y)$

$\frac{cosecA+cotA-(cosecA+cotA)(cosecA-cotA)}{(cotA-cosecA+1)}$

taking common

$\frac{(cosecA+cotA)(1-cosecA+cotA)}{(cotA-cosecA+1)}$

cancelling like terms in numerator and denominator

we are left with $cosecA+cotA=\frac{1}{sinA}+\frac{cosA}{sinA}=\frac{(1+cosA)}{sinA}$