Find the value sin4θ - cos4θ + 2cos2θ, when θ=Π7
Before we actually solve this, let's find out the value of this expression for some known angles for us.
θ = 0
F(0) = sin4θ - cos4θ + 2cos2θ = 0 - 1 + 2 = 1
θ=Π2
F (Π2) = 1 - 0 + 2 × 0 = 1
θ = Π4
F (Π2) = 14 - 14 + 2 × 12 = 1
We see that the value of the expression remains the same for different θ, we considered. The motivation for doing this is θ = θ = Π7 is not a standard / commonly used angle. So the assumption is that the expression will be independent of θ (We can't be sure about this)
We have to simplify / modify the expression. One way of proceeding is by replacing sin2θ or cos2θ with 1 - cos2θ or 1 - sin2θ; another way of proceeding is by taking cos2θ from the last two terms. If we proceed this was, it won't simplify the expression a lot. So we will go with replacing sin2θ with 1 - cos2θ.
⇒ sin4θ - cos4θ + 2cos2θ = (1−cos2θ)2 - cos4θ + 2 cos2θ
= 1 + cos4θ - 2 cos2θ - cos4θ + 2 cos2θ
= 1
As we expected, the expression is independent of θ.
Key steps:
(1) Guessing that the expression could be independent of θ
(2) Attempting to modify the expression using basic identities (cos2θ + sin2θ = 1)