How would I simplify (1-cosx/tanx)+ (sinx/1+cosx) using trigonometric identities?

Question

The answer is sinx but I want to know the steps in getting that answer.

Answer 1

[(1-cosx)/tanx]+[sinx/(1+cosx)]

 

we know that tanx=sinx/cosx, substituting that in the above equation we have

[(1-cosx)/(sinx/cosx)] +sinx/(1+cosx)

 

Now, multiply the numerator and denominator with cosx for the first term

 

=[(1-cosx)cosx/sinx] +sinx/(1+cosx)

 

Now we add the 2 terms

 

{[(1-cosx)cosx](1+cosx)+sinx.sinx}/sinx.(1+cosx)  ------------(1)

 

reordering the first term:

cosx(1-cosx)(1+cosx)

=cosx(1-cos²x)                                         (since 1-cos²x =sin²x)

=cosx. sin²x

 

by substituting this value in (1)

{cosx.sin²x+sin²x}/sinx.(1+cosx)

⁼sin²x.(cosx+1)/sinx.(1+cosx)

 

so the result is sinx.

 

Please let me know if you have questions