How to verify this identity? (sin 2x / sinx) - (cos 2x / cos x) = sec x   thank you!

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sin2x/sinx - cos2x/cosx = secx

We will start from the L.H.S and prove the identity.

We know that:

 sin2x = 2sinx*cosx

cos2x = 2cos^2 x - 1

We will substitute in L.H.S.

=> 2sinxcosx/sinx  - (2cos^2 x-1)/cosx

==> 2cosx - 2cos^2 x/cosx  + 1/cosx

==> 2cosx - 2cosx + 1/cosx

Reduce 2cosx

Now we know that secx = 1/cosx

==> 1/cosx = sec x.........R.H.S

Then the identity "(sin 2x / sinx) - (cos 2x / cos x) = sec x" is TRUE.

 

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