Perform the mlutiplication and use the fundamental identiteis to simplify.
(2csc x +2)(2csc x -2)
answer: 4cot^2x
But HOW???
Multiply each thing in the first set of parenthesees by each thing in the second set of parenthesees and add the products together.
(2csc x + 2)(2csc x - 2)
= (2csc x)(2csc x) + (2csc x)(-2) + (2)(2csc x) + (2)(-2)
= 4csc2x - 4csc x + 4 csc x - 4
= 4csc2x - 4
Factor out a 4.
= 4(csc2x - 1)
Rewrite like this.
\(=4(\frac{1}{\sin^2x}-1)=4(\frac{1}{\sin^2x}-\frac{\sin^2x}{\sin^2x})=4(\frac{1-\sin^2x}{\sin^2x})\)
Then, use the Pythagorean identity that says sin2 + cos2 = 1
\(=4(\frac{cos^2x}{sin^2x})\)
= 4cot2x
Multiply each thing in the first set of parenthesees by each thing in the second set of parenthesees and add the products together.
(2csc x + 2)(2csc x - 2)
= (2csc x)(2csc x) + (2csc x)(-2) + (2)(2csc x) + (2)(-2)
= 4csc2x - 4csc x + 4 csc x - 4
= 4csc2x - 4
Factor out a 4.
= 4(csc2x - 1)
Rewrite like this.
\(=4(\frac{1}{\sin^2x}-1)=4(\frac{1}{\sin^2x}-\frac{\sin^2x}{\sin^2x})=4(\frac{1-\sin^2x}{\sin^2x})\)
Then, use the Pythagorean identity that says sin2 + cos2 = 1
\(=4(\frac{cos^2x}{sin^2x})\)
= 4cot2x