Question

1 + Cos A upon sin a + sin a upon 1 + cos A is equals to 2 cosec a

Answer 1

Given equation is 1 + cos A/sin A + sin a/1 + cos A

On cross-multiplication, we get

                             = sin^2 A + (1 + cos A)^2/sin A(1 + cos A)

 We know that (a + b)^2 = a^2 + b^2 + 2ab

                              = sin^2 A + (1 + cos^2 A + 2 cos A)sinA(1 + cos A)

                              = sin^2 A + 1 + cos^2 A + 2 cos A)/sin A(1 + cos A)

We know that sin^2 theta + cos^2 theta = 1

                              = sin^2 A + cos^2 A + 1 + 2 cos A/sin A(1 + cos A)

                              = 1 + 1 + 2 cos A/sin A(1 + cos A)

                              = 2 + 2 cos A/sin A(1 + cos A)

                              = 2(1 + cos A)/sin A(1 + cos A)

                              = 2/sin A

We know that 1/sin theta = cosec theta

                              = 2 cosec A.


LHS = RHS.

Hope this helps!