prove that cos4 theta - cos2 theta = sin4 theta sin2 theta
Answers
CORRECT QUESTION:
Prove that cos⁴ θ - cos² θ = sin⁴ θ - sin² θ
GIVEN:
- cos⁴ θ - cos² θ = sin⁴ θ - sin² θ
TO PROVE:
- cos⁴ θ - cos² θ = sin⁴ - θ sin² θ
EXPLANATION:
METHOD 1:
By taking L.H.S as cos⁴ θ - cos² θ
Take cos² θ as common
cos² θ (cos² θ - 1)
cos² θ ( - sin² θ )
(1 - sin² θ )( - sin² θ )
- sin² θ + sin⁴ θ
sin⁴ θ - sin² θ
HENCE PROVED.
METHOD 2:
By taking R.H.S as sin⁴ θ - sin² θ
Take sin² θ as common
sin² θ (sin² θ - 1)
sin² θ ( - cos² θ )
(1 - cos² θ )( - cos² θ)
- cos² θ + cos⁴ θ
cos⁴ θ - cos² θ
HENCE PROVED.
Answer:
Prove that cos⁴ θ - cos² θ = sin⁴ θ - sin² θ
GIVEN:
cos⁴ θ - cos² θ = sin⁴ θ - sin² θ
TO PROVE:
cos⁴ θ - cos² θ = sin⁴ - θ sin² θ
EXPLANATION:
METHOD 1:
By taking L.H.S as cos⁴ θ - cos² θ
Take cos² θ as common
cos² θ (cos² θ - 1)
\boxed{\large{\bold{1-\sin^2 \theta = \cos^2\theta}}}
1−sin
2
θ=cos
2
θ
\boxed{\large{\bold{-\sin^2 \theta = \cos^2\theta-1}}}
−sin
2
θ=cos
2
θ−1
cos² θ ( - sin² θ )
(1 - sin² θ )( - sin² θ )
- sin² θ + sin⁴ θ
sin⁴ θ - sin² θ
HENCE PROVED.
METHOD 2:
By taking R.H.S as sin⁴ θ - sin² θ
Take sin² θ as common
sin² θ (sin² θ - 1)
\boxed{\large{\bold{1-\cos^2 \theta = \sin^2\theta}}}
1−cos
2
θ=sin
2
θ
\boxed{\large{\bold{-\cos^2 \theta = \sin^2\theta}-1}}
−cos
2
θ=sin
2
θ−1
sin² θ ( - cos² θ )
(1 - cos² θ )( - cos² θ)
- cos² θ + cos⁴ θ
cos⁴ θ - cos² θ
HENCE PROVED