Question

Prove that): (1 + 3cos theta - 4cos^3 theta)/(1 - cos theta) = (1 + 2cos theta) ^ 2

Answer 1

To prove the given identity, we simplify the left-hand side and the right-hand side of the equation to show that they are equal.

Step-by-step explanation:

To prove the given identity:

(1 + 3cos(theta) - 4cos^3(theta))/(1 - cos(theta)) = (1 + 2cos(theta))^2

We can start by simplifying the left-hand side of the equation:

=(1 + 3cos(theta) - 4cos^3(theta))/(1 - cos(theta))

=(1 + cos(theta)(3 - 4cos^2(theta)))/(1 - cos(theta))

=(1 + cos(theta)(3 - 2cos^2(theta) - 2cos^2(theta)))/(1 - cos(theta))

=(1 + cos(theta)(3 - 2cos^2(theta) - sin^2(theta)))/(1 - cos(theta))

=(1 + cos(theta)(3 - sin^2(theta) - 2cos^2(theta)))/(1 - cos(theta))

=(1 + 3cos(theta) - cos^2(theta) - sin^2(theta))/(1 - cos(theta))

=(1 + 3cos(theta) - (cos^2(theta) + sin^2(theta)))/(1 - cos(theta))

=(1 + 3cos(theta) - 1)/(1 - cos(theta))

=3cos(theta)/(1 - cos(theta))

Now, we can simplify the right-hand side of the equation:

(1 + 2cos(theta))^2

= (1 + 2cos(theta))(1 + 2cos(theta))

= 1 + 2cos(theta) + 2cos(theta) + 4cos^2(theta)

= 1 + 4cos(theta) + 4cos^2(theta)

= 4cos^2(theta) + 4cos(theta) + 1

So, the right-hand side is equal to 3cos(theta)/(1 - cos(theta)) as well.

Therefore, the given identity is proven.