The directions say to prove the identity. I understand how these two functions work, but I don't understand how to go through the process of getting from the left side to the right side.
When trying to prove trig identities, it is often helpful to convert TAN functions into SIN/COS functions:
Proof Step 1: Start with the original equation to prove:
tan2x - sin2x = (tan2x)(sin2x)
Proof Step 2: Replace tan with sin/cos
(sin2x/cos2x) - sin2x = (sin2x/cos2x)(sin2x)
Proof Step 3: Obtain a common denominator on left, simplify right
(sin2x - sin2x cos2x) / cos2x = sin4x / cos2x
Proof Step 4: Cancel cos2x from both denominators
sin2x - sin2x cos2x = sin4x
Proof Step 5: Factor out sin2x from left
(sin2x)(1 - cos2x) = sin4x
Proof Step 6: Use trig identity 1-cos2x = sin2x
(sin2x)(sin2x) = sin4x
Proof Step 7: Simplify, DONE.
sin4x = sin4x