Verify that: 1. (CosX)(CotX) = CscX - SinX

2. (Csc X - CotX)^2 = 1-CosX/1+CosX

These were the last two questions on my Homework that I couldn’t figure out after many tries! Thank you to anyone who helps!

See Below.

Step-by-step explanation:

Problem 1)

We want to verify that:

Note that cot(x) = cos(x) / sin(x). Hence:

Multiply:

Recall that Pythagorean Identity: sin²(x) + cos²(x) = 1 or cos²(x) = 1 - sin²(x). Substitute:

Split:

Simplify:

Problem 2)

We want to verify that:

Square:

Convert csc(x) to 1 / sin(x) and cot(x) to cos(x) / sin(x). Thus:

Factor out the sin²(x) from the denominator:

Factor (perfect square trinomial):

Using the Pythagorean Identity, we know that sin²(x) = 1 - cos²(x). Hence:

Factor (difference of two squares):

Factor out a negative from the first factor in the denominator:

Cancel:

Distribute the negative into the numerator. Therefore: