Question

Verify the identity. cos 4x + cos 2x = 2 - 2 sin2 2x - 2 sin2 x

Answer 1

cos 4x + cos 2x = 2 - 2 sin^2 2x - 2 sin^2 x
cos 4x + cos 2x = 1 + 1 − 2sin^2(2x) − 2sin^2(x)
 cos 4x + cos 2x = (1 - 2sin^2(2x)) + (1 - 2 sin^2(x)
 cos 4x + cos 2x = cos 4x + cos 2x

Answer 2

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