Question

Which expression is equivalent to cot2β(1−cos2β) for all values of β for which cot2β(1−cos2β) is defined?

OPTIONS:
Select the correct answer below:
a) cot2β
b) 1
c) secβtanβ
d) sec3β
e) cos2β

Answer 1

Option D: sec3β

Explanation:

To find an equivalent expression to cot2β(1−cos2β), we can use trigonometric identities to simplify the expression. Here’s how:

Use the identity cos(2θ) = 1 - 2sin²(θ) to rewrite cos²(β) as (1 - cos(2β))/2.

Use the identity cot(θ) = cos(θ)/sin(θ) to rewrite cot²(β) as cos²(β)/sin²(β).

Substitute the expressions from steps 1 and 2 into cot²(β)(1 - cos²(β)) and simplify.

Here’s what that looks like:

cot²(β)(1 - cos²(β) = (cos²(β)/sin²(β)) * (1 - (1 - cos(2β))/2) = (cos²(β)/sin²(β)) * (cos(2β)/2) = (cos²(β) * cos(2β))/(2sin²(β))

We can simplify this expression further using the identity sin²(θ) + cos²(θ) = 1 to get:

(cot²(β)(1 - cos²(β)) / sin²(β) = (cos²(β) * cos(2β))/(2sin⁴(β)) = (cos²(β) * 2cos²(β) - 1)/(2sin⁴(β)) = (2cos⁴(β) - cos²(β))/(2sin⁴(β))

Therefore, the equivalent expression to cot2β(1−cos2β) is:

I hope this helps