3. Find the value of θ cos4(θ) + sin2(θ) = cos2(θ) + sin4(θ)

Question

they are squared and to the power of 4 

Answer 1

Start by moving the terms around:

 

cos⁴θ - sin⁴θ = cos²θ - sin²θ

 

now you have difference of perfect squares on both sides, so factor both:

 

(cos²θ - sin²θ)(cos²θ + sin²θ) = (cosθ - sinθ)(cosθ + sinθ)

 

now on the left you have cos²θ + sin²θ which is equal to 1

and you have difference of perfect squares that you can factor again:

 

(cosθ - sinθ)(cosθ + sinθ) * 1 = (cosθ - sinθ)(cosθ + sinθ)

 

since both sides are identical, θ is equal to all real numbers (test out any value and see for yourself!)