Solve for Θ for 0 is less than or equal to Θ less than or equal to 180°. 8 cos^4 Θ = cos 4Θ + 4cos 2Θ +3?

Question

A) The equation is an identity( every Θ is a solution) 
B) 35°, 90°, 145° 
C) 0° 
D) 0°, 360°

 

Answer 1

The trigonometric equation is 8cos⁴ (θ) = cos(4θ) + 4 cos(2θ) + 3, where 0 ≤ θ ≤ π.

8cos⁴ (θ) = cos(2 * 2θ) + 4 cos(2θ) + 3

Double angle formula : cos(2θ) = 2cos² θ - 1.

8cos⁴ (θ) = 2cos² (2θ) - 1 + 4(2cos² θ - 1) + 3

8cos⁴ (θ) = 2(2cos² θ - 1)² - 1 + 8cos² θ - 4 + 3

8cos⁴ (θ) = 2(4cos⁴ θ - 4cos² θ + 1)² - 2 + 8cos² θ

8cos⁴ (θ) = 8cos⁴ θ - 8cos² θ + 2 - 2 + 8cos² θ

8cos⁴ (θ) = 8cos⁴ θ - 8cos² θ + 2 - 2 + 8cos² θ

8cos⁴ (θ) = 8cos⁴ θ.

LHS = RHS.

So, the equation is an identity.

Since, the given equation is an identity, every θ is a solution.

 

Option A is the correct choice.