Question

Question #0e194

Answer 1

First, change all trigonometric functions to `sin(x)` and `cos(x)` by using the identities `csc(x)=1/sin(x)` and `cot(x)=1/tan(x)=1/(sin(x)/cos(x))=cos(x)/sin(x)`.

This results in `sin(x)/(1-cos(x))=1/sin(x)+cos/sin(x)`. Add the two fractions on the right-hand side: `sin(x)/(1-cos(x))=(1+cos(x))/sin(x)`.

Now, multiply both sides by `sin(x)(1-cos(x))` to eliminate the fractions. The equation becomes `sin^2(x)=(1+cos(x))(1-cos(x))`. Expand the right-hand side by using the identity `(a+b)(a-b)=a^2-b^2`.

This results in `sin^2(x)=1-cos^2(x)`. Add `cos^2(x)` to both sides to get `sin^2(x)+cos^2(x)=1`, which is one of the trigonometry identites.