Question

Prove that : (Sin Ө + Cosec Ө)/(Tan Ө+Cot Ө) = Sin Ө + Cos Ө

Answer 1

Use the FOIL method.
Make sure that you write each more complex function in their sin/cos components. You should be able to cancel a lot and wind up with only sin theta plus cos theta.

Answer 2

I will do it in parts for easier typing

sinØ + cscØ
= sinØ + 1/cosØ
= (sinØcosØ + 1)/cosØ

tanØ + cotØ
= sinØ/cosØ + cosØ/sinØ
= (sin^2 Ø + cos^ Ø)/sinØcosØ
= 1/(sinØcosØ

LS = (Sin Ө + Cosec Ө)/(Tan Ө+Cot Ө)
= (sinØcosØ + 1)/cosØ ÷ 1/(sinØcosØ
= ((sinØcosØ + 1)/cosØ)x( sinØcosØ)/1
= (sinØcosØ + 1)(sinØ)
= (sin^2 Ø)(cosØ) + sinØ
≠ RS

I tested it for Ø = 20° and the original equation does not satisfy.
All you need is one exception and your equation is NOT an identity. Yours is not.

I usually test at the beginning, could have saved myself a lot of typing.