How do you prove this identity? sec(x) − sin(x)tan(x)= cos(x)

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Answer 1 · 2018-02-25T04:23:35

Rewrite in terms of #sin(x)# and #cos(x)#, subtract, apply the identity #1-sin^2(x)=cos^2(x)#, and simplify.

Recall that

#sec(x)=1/cos(x)# and #tan(x)=sin(x)/cos(x)#

Rewrite the identity in terms of #sin(x)# and #cos(x):#

#1/cos(x)-sin(x)sin(x)/cos(x)=cos(x)#

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

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

Recall that

#sin^2(x)+cos^2(x)=1#

Rewriting this a little, we see that

#1-sin^2(x)=cos^2(x)#

Replacing #1-sin^2(x)# with #cos^2(x)# in the numerator of our expression, we get

#cos^2(x)/cos(x)=cos(x)#

#(cos(x)cancelcos(x))/cancelcos(x)=cos(x)#

#cos(x)=cos(x)#