Evaluate #sec(x)(sec(x)+tan(x))=1/(1-sin(x))#?

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Answer 1 · 2018-04-17T14:25:03

It is an identity; true for all real values of x.

Prove: #sec(x)(sec(x)+tan(x))=1/(1-sin(x))#

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

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

Combine the fractions with the common denominator:

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

Perform the multiplication:

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

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

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

Factor the denominator:

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

Please observe that #(1+sin(x))/(1+sin(x)) to 1#:

#1/(1-sin(x))=1/(1-sin(x))# Q.E.D.