Question

If cosec $\theta =2$, show that $(cot\theta +\frac{sin\theta }{1+cos\theta })$ = 2.

Answer 1

cosecθ=2
to show that
{(cotθ+sinθ)/(1+cosθ)}=2
cosecθ=2
→1/sinθ=2 i.e sinθ=1/2
and cosθ=√(1−sin²θ)= √{1−(1/2)²)=√3/2
LHS ={(cosθ/sinθ)+sinθ/(1+cosθ)}
[{(√3/2)/(1/2)+(1/2)/(1+√3/2)}]
=[{√3}+{(1/2)/(1+√3/2)}]
=[{√3}+1/{(2+√3)}]
={√3+1/(2+√3)}
={√3+(2−√3)}/{(2+√3)(2−√3)}
={√3+2−√3)}/(4−3)}
={2)/(1)
=2=RHS