Question

if cos alpha + isinalpha =cos(theta+itheta) then prove that Sinsquare theta =plus or minus sin alpha

Answer 1

Given:cos(θ−α),cosθ,cos(θ+α) are in H.P

⇒

cos(θ−α)

1

,

cosθ

1

,

cos(θ+α)

1

are in A.P

cosθ

1

−

cos(θ−α)

1

=

cos(θ+α)

1

−

cosθ

1

cosθ

2

=

cos(θ+α)

1

+

cos(θ−α)

1

⇒

cosθ

2

=

cos(θ+α)cos(θ−α)

cos(θ−α)+cos(θ+α)

⇒

cosθ

2

=

cos

2

θ−sin

2

α

cosθcosα−sinθsinα+cosθcosα+sinθsinα

⇒

cosθ

2

=

cos

2

θ−sin

2

α

2cosθcosα

⇒

cosθ

1

=

cos

2

θ−sin

2

α

cosθcosα

⇒cos

2

θ−sin

2

α=cos

2

θcosα

⇒sin

2

α=cos

2

θ−cos

2

θcosα

⇒sin

2

α=cos

2

θ(1−cosα)

⇒

(1−cosα)

sin

2

α

=cos

2

θ

⇒

(1−cosα)

1−cos

2

α

=cos

2

θ

⇒

(1−cosα)

(1−cosα)(1+cosα)

=cos

2

θ

⇒1+cosα=cos

2

θ

Hence cos

2

θ=1+cosα

Hence proved.

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