How do I simplify this expression using a trig. identity

Question

Use a trigonometric identity to simplify the expression.

sin∝ (csc∝-sin∝)

Answer 1

First, let's work with the latter half of the expression csc(θ) - sin(θ):

csc(θ) = 1/sin(θ) so we can write the latter portion [csc(θ) - sin(θ)] as [1/sin(θ) - sin(θ)/1]. These are two fractions that require a common denominator to add. So, multiply sin(θ)/1 by sin(θ)/sin(θ) to give you:

[1/sin(θ) - sin²(θ)/sin(θ)] or

[1 - sin²(θ)]/sin(θ)

Now, use the Pythagorean Identity sin²(θ) + cos²(θ) = 1 and rearrange it to say cos²(θ) = 1 - sin²(θ)

Replace the "1 - sin²(θ)" with cos²(θ) to get:

cos²(θ)/sin(θ)

Now, combine that with the first half of the expression to get:

sin(θ)•cos²(θ)/sin(θ) and the sin(θ)'s cancel out leaving the expression equal to cos²(θ)

Answer 2

sina(csca -sina) distribute gives sinnacsca -sin^2 a Substitute 1/sin for csc

sina (1/sina) -sin^2 a

1- sin^2 a which is cos^2a

Final answer is cos^2 a