Maths - Any Ideas?

Question

Hello,

There are some questions that I am not sure how to tackle:

Prove the following identities:

1) sin(θ + 90) = cosθ

2) sin(A + B) + sin(A - B) = 2sinA cosB

3) cosx cos2x + sinx sin2x = cos x

Thank you in advance.

Answer 1

1) sin(θ + 90) = cosθ

Take left hand identity:

sin(θ + 90)=

Using angle sum and difference identities:

sin(A+B)=sinAcosB+cosAsinB

A=θ and B =90

sin(θ + 90) =sinθcos90° +cosθsin90°

Here sin90° =1 and cos 90° =0

 =sinθ(0) +cosθ(1)

 =cosθ

  =Right hand identity:

----------------------------------

2) sin(A + B) + sin(A - B) = 2sinA cosB

Take left hand identity:

sin(A + B) + sin(A - B)

Using angle sum and difference identities:

sin(A+B)=sinAcosB+cosAsinB

sin(A-B)=sinAcosB-cosAsinB

sin(A + B) + sin(A - B) =(sinAcosB+cosAsinB)+(sinAcosB-cosAsinB)

                                       = 2sinAcosB   =Right hand identity:

 

 

Answer 2

3)

cosx cos2x + sinx sin2x = cos x

Using angle sum and difference identities:

cos(A-B)=cosAcosB+sinAsinB

A  = 2x ,B=x

cosx cos2x + sinx sin2x = cos x

=cos(2x-x)

=cozx