Need help proving this trig identity!!

Question

1/sec(x)tan(x)=csc(x)-sin(x)

Answer 1

1/sec(x)tan(x)=csc(x)-sin(x)

Working with the right side of the equation :

Step 1: Replace csc(x) with 1/sin(x) to get 1/sin(x) – sin(x)

Step 2: Get a common denominator (use sin(x)) to get: 1/sin(x) – sin^2(x)/sin(x)

Step 3: Combine numerators to get (1-sin^2(x))/sin(x)

Step 4: Using the Pythagorean Identity sin^2(x) + cos^2(x) = 1, modify to by subtracting sin^2(x) from both sides to become cos^2(x) = 1 – sin^2(x) and replace the numerator with cos^2(x) to get cos^2(x)/sin(x)

Step 5: Rewrite the fraction as 1/(sin(x)/cos^2(x))

Step 6: Separate the cosines to get: 1/((sin(x)/cos(x))*(1/cos(x)))

Step 7: Rewrite sin(x)/cos(x) as tan(x) and 1/cos(x) as sec(x) to get 1/(sec(x)tan(x))