Question

Prove that `(cos3 theta)/(cos theta)+(sin3 theta)/(sin theta)=4cos2 theta`​

Answer 1

Step-by-step explanation:

LHS:

cosθ−sinθ

cos

3

θ−sin

3

θ

+

cosθ+sinθ

cos

3

θ+sin

3

θ

we use two identities,

⇒a

3

−b

3

=(a−b)(a

2

+b

2

+ab)

⇒a

3

+b

3

=(a+b)(a

2

+b

2

−ab)

using the above identities,

=

(cosθ−sinθ)

(cosθ−sinθ)(cos

2

θ−sin

2

θ+cosθ.sinθ)

+

(cosθ+sinθ)

(cosθ+sinθ)(cos

2

θ−sin

2

θ−cosθ.sinθ)

=cos

2

θ+sin

2

θ+cosθ.sinθ+cos

2

θ+sin

2

θ−cosθ.sinθ

=2(cos

2

θ+sin

2

θ)

=2 (∵cos

2

θ+sin

2

θ=1)

Hence, proved.