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

Question

13102 views around the world

You can reuse this answer
Creative Commons License

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):

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

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

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

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

$color(white)(RHS)=LHS$

How do you verify 1-tan^2x=2-sec^2x1-tan^2x=2-sec^2x?