sin theta + sin 3 theta = tan theta sec theta

Question

sin theta + sin 3 theta = tan theta sec theta 

Show that the right side equals the left side

Answer 1

For sinθ+sin3θ=tanθsecθ

left side 2sin((θ+3θ)/2)cos((θ-3θ)/2)=2sin2θcosθ  (cosine is an even function)

=2sinθcosθcosθ=2sinθcos²θ

For the right side tanθsecθ=tanθ/cosθ=sinθ/cos²θ

These two expressions are equal only when sinθ=0 or cos^4_θ=1/2

I tried this with θ=25° and got

0.436332313 for the right side
1.388544088 for the left side

Answer 2

The left side is NOT equivalent to the right side.

 

For example, if θ = π/3, then  sinθ + sin(3θ) = sin(π/3) + sinπ

                                                                

                                                                              = √3/2 + 0 = √3/2

 

                                                  but, tanθsecθ = tan(π/3)sec(π/3)

 

                                                                           = (√3)(2) = 2√3