Question

1)cot a/2 -tan a/2 = 2 cot a2) cot b/2 + tan b/2= 2 cosec b prove​

Answer 1

1)

LHS = cot(a/2) - tan(a/2)

= (1 - tan^2(a/2))/tan(a/2)

= (2-sec^2(a/2))/tan(a/2)

= 2cot(a/2) - cosec(a/2)sec(a/2)

= 2(1+cos(a))/sin(a) - 1/(cos(a/2)sin(a/2))

= 2 (1+cos(a))/sin(a) - 2/sin(a)) (product to sums)

= 2[(1+cos(a) -1)/sin(a)]

=2cot a

= RHS

2.

LHS = cot(b/2) + tan(b/2)

= [1 + tan^2(b/2)]/tan(b/2)

= sec^2(b/2)/tan(b/2)

= 1/sin(b/2)cos(b/2)

using product to sums

= 2/sin(b)

= 2cosec(b)

= RHS