Prove that sinθ+sin3θ+sin5θ+sin7θ/cosθ+cos3θ+cos5θ+cos7θ=tan4θ

Steps to prove sinθ+sin3θ+sin5θ+sin7θ/cosθ+cos3θ+cos5θ+cos7θ=tan4θ

Take L.H.S

=sinθ+sin3θ+sin5θ+sin7θ/cosθ+cos3θ+cos5θ+cos7θ

=sinθ+sin7θ+sin3θ+sin5θ/cosθ+cos7θ+cos3θ+cos5θ

=         {2.sin(θ+7θ)/2.cos(θ-7θ)/2+2.sin(3θ+5θ)/2.cos(3θ-5θ)/2}/{2.cos(θ+7θ)/2.cos(θ-7θ)/2+2.cos(3θ+5θ)/2.cos(3θ-5θ)/2}

={2.sin4θ.cos(-3θ)+2.sin4θ.cos(-θ)}/{2.cos4θ.cos(-3θ)+2.cos4θ.cos(-θ)}

=2sin4θ{cos(-3θ)+cos(-θ)}/2cos4θ{cos(-3θ)+cos(-θ)}

=sin4θ/cos4θ

=tan4θ

=R.H.S

Hence sinθ+sin3θ+sin5θ+sin7θ/cosθ+cos3θ+cos5θ+cos7θ=tan4θ is proved.

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