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# Perform the mlutiplication and use the fundamental identiteis to simplify.

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Perform the mlutiplication and use the fundamental identiteis to simplify.

(2csc x +2)(2csc x -2)

But HOW???

Guest Apr 23, 2017

#1
+7324
+3

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

hectictar  Apr 23, 2017
#1
+7324
+3

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

hectictar  Apr 23, 2017