prove:4(Cot-13+cosec-1√5)=π

Question

how to prove it?

Answer 1

4(cot^-1(3)+cosec^-1(5))=4(tan^-1(1/3)+sin^-1(1/√5)).

Let x=tan^-1(1/3) and y=sin^-1(1/√5), then:

tan(x)=1/3, so sin(x)=1/√10, cos(x)=3/√10, and sin(y)=1/√5, cos(y)=2/√5

sin(x+y)=sin(x)cos(y)+cos(x)sin(y)=

2/√50+3/√50=5/√50=√50/10=5√2/10=√2/2=sin(π/4).

Therefore sin(x+y)=sin(tan^-1(1/3)+sin^-1(1/√5))=sin(π/4).

So, tan^-1(1/3)+sin^-1(1/√5)=π/4, and 4(tan^-1(1/3)+sin^-1(1/√5))=π QED