How do you verify the identity #(tanx+secx)(1-sinx)=cosx#?

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Answer 1 · 2016-08-26T20:06:03

See below.

Use the following identities to simplify the left-hand side:

•#tanbeta = sinbeta/cosbeta#

•#secbeta = 1/cosbeta#

Start the simplification process:

#(sinx/cosx + 1/cosx)(1 - sinx) = cosx#

#((sinx + 1)/(cosx))(1 - sinx) = cosx#

#(1 - sin^2x)/cosx = cosx#

Now, rearrange the pythagorean identity #sin^2beta + cos^2beta = 1# for #cos^2beta# to get #cos^2beta = 1- sin^2beta#. Applying this to our problem:

#cos^2x/cosx = cosx#

#((cosx)(cosx))/cosx = cosx#

#(cancel(cosx)cosx)/cancel(cosx) = cosx#

#cosx = cosx#

Identity proved!!

Hopefully this helps!