Question

Consider the proof for the identity csc(x)+sin(x)=(2-cos^(2)(x))/(sin(x)).

Answer 1

1. given:

csc(x) + sin(x)

2. definition of csc(x) :

1/sin(x) + sin(x)

3. combining fractions with a common denominator:

(1 + sin²(x))/sin(x)

4. expanding 1 = cos²(x) + sin²(x) :

(cos²(x) + sin²(x) + sin²(x))/sin(x)

5. simplifying:

(2 sin²(x) + cos²(x))/sin(x)

6. employing the same identity as in (4) :

(2 (1 - cos²(x)) + cos²(x))/sin(x)

7. expanding 2 (1 - cos²(x)) :

(2 - 2 cos²(x) + cos²(x))/sin(x)

8. simplifying:

(2 - cos²(x))/sin(x)