Question #3a93e

2 Answers
May 7, 2017

Use Pythagorean identity and definition of tangent

Explanation:

tan^2(x)+1 = 1/cos^2(x)

Remember, tan(x)=sin(x)/cos(x) (this is just the definition of tan(x))

tan^2(x)+1

=(tan(x))^2+1

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

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

Common denominator

=sin^2(x)/cos^2(x)+cos^2(x)/cos^2(x)

=(sin^2(x)+cos^2(x))/cos^2(x)

Now we use the Pythagorean identity (1=cos^2(x)+sin^2(x), I won't prove this here, but if you want, check this out) to solve the rest

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

There it is,

tan^2(x)+1=1/cos^2(x)

May 7, 2017

We know by definition
costheta=x/r
tantheta=y/x
enter image source here
Now,
1+tan^2theta=1+y^2/x^2
=(x^2+y^2)/x^2
=r^2/x^x
=1/cos^2theta