Answer:
The equation Cos4x-cos2x = 2 - 2sin²2x - 2sin²x is TRUE
Step-by-step explanation:
To simplify the trigonometry identity
cos 4x + cos 2x, we use the double angle formula .
According to double angle formula,
Cos2A = cos(A+A)
= CosAcosA-SinASinA
= cos²A - Sin²A
= (1-sin²A)-Sin²A
= 1-2sin²A
Similarly, Cos4x = cos(2x+2x)
= cos²2x - sin²2x... (1)
Since cos²2x+Sin²2x = 1
Cos²2x = 1-sin²2x...(2)
Substituting eqn 2 in 1, we have;
Cos4x= (1-sin²2x) - sin²2x
Cos4x = 1 - 2sin²2x... (3)
Similarly for Cos2x;
Cos2x = cos²x-sin²x
Cos2x = (1-sin²x)-sin²x
Cos2x = 1 - 2sin²x...(4)
Cos4x + cos2x = 1 - 2sin²2x +(1-2sin²x)
= 1 - 2sin²2x + 1 - 2sin²x
= 2 - 2sin²2x - 2sin²x
Which proves the equation;
Cos4x-cos2x = 2 - 2sin²2x - 2sin²x
