Prove te folowing identity is true:

Question

   sin x        1 + cos x

--------- + --------------=2 csc x

1+cos x        sin x

Answer 1

to prove this is true, you can only work with one side at a time, I choose to work with the left. To add the two fractions we need to bring them to a common denominator.

The Least common denominator is (1+cosx)(sinx). Therefore, the first fraction needs to be multiplied by sin(x), becoming sin^2(x)/(sinx)(1+cosx).

To bring the second fraction to the same denominator, we need to multiply it by (1+cosx). Thus, the second fraction becomes (1 +2cosx+ cos^2(x))/(sinx)(1+cosx).

 

When we add them together, we get sin^2(x)+1+2cos(x)+cos^2(x) /(sinx)(1+cosx).

 

Adding like terms and simplifying the numerator results in 2+2cosx, which can then be factored as 2(1+cosx)/(sinx)(1+cosx). We cross out the (1+cosx) from the numerator and the denominator, leaving us with 2/sinx, which is the same as 2cscx