Prove that cos7α+cos3α-cos5α-cosα/sin7α-sin3α-sin5α+sinα=cot2α

Steps to prove cos7α+cos3α-cos5α-cosα/sin7α-sin3α-sin5α+sinα=cot2α

Take L.H.S

=cos7α+cos3α-cos5α-cosα/sin7α-sin3α-sin5α+sinα

=cos7α-cosα+cos3α-cos5α/sin7α-sinα-(sin3α+sin5α)

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

=(-2.sin4α.sin3α+2.sin4α.sinα)/(2.sin4α.cos3α-2.sin4α.cosα)

=(sin3α-sinα)/(cosα-cos3α)

={2.cos(3α+α)/2.sin(3α-α)/2}/{2.sin(3α+α)/2.sin(3α-α)/2}

=cos2α/sin2α

=cot2α

=R.H.S

Hence cos7α+cos3α-cos5α-cosα/sin7α-sin3α-sin5α+sinα=cot2α is proved.

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