Prove the identity

Question

(Cosxsinx) ÷ tanx = 1 - sin²x

Answer 1

To prove the identity, we will start with the left-hand side (LHS) and manipulate it to obtain the right-hand side (RHS).

LHS: (cos(x)sin(x))/tan(x)

Recall that tan(x) = sin(x)/cos(x). Substituting this in the LHS, we get:

LHS: (cos(x)sin(x))/(sin(x)/cos(x))

Now, let's simplify:

LHS: (cos(x)sin(x)) * (cos(x)/sin(x))

Cancel out sin(x) from the numerator and denominator:

LHS: cos²(x)

Recall the Pythagorean identity: sin²(x) + cos²(x) = 1. Solve for cos²(x) by subtracting sin²(x) from both sides:

cos²(x) = 1 - sin²(x)

Therefore, we have:

LHS = 1 - sin²(x)

which is the same as the RHS.

Hence, we have proved the identity:

(cos(x)sin(x))/tan(x) = 1 - sin²(x)