Question #3a93e

2 Answers
C. McCracken
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)#

Deepak Pandey
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#

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