How do you verify #1-tan^2x=2-sec^2x#?

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Answer 1 · 2018-03-05T23:05:10

Use the Pythagorean identity:

#color(white)=>sin^2x+cos^2x=1#

Then, divide all the terms by #cos^2x# to derive a new identity:

#color(white)=>sin^2x/cos^2x+cos^2x/cos^2x=1/cos^2x#

#color(white)=>tan^2x+1=sec^2x#

Now, here's the actual proof (starting with the right side):

#RHS=2-color(red)(sec^2x)#

#color(white)(RHS)=2-(color(red)(1+tan^2x))#

#color(white)(RHS)=2-1-tan^2x#

#color(white)(RHS)=1-tan^2x#

#color(white)(RHS)=LHS#