Please how I can prove that ? Cos^2(t)=1/1+tan^2(t) Thanks

2 Answers
Mar 22, 2018

I think you mean "prove" not "improve". See below

Explanation:

Consider the RHS

#1/( 1+ tan^2(t))#

#tan(t) = sin(t)/cos(t)#

So, #tan^2(t) = sin^2(t)/cos^2(t)#

So RHS is now:
#1/( 1+ (sin^2(t)/cos^2(t))#

#1/(( cos^2(t)+ sin^2(t))/cos^2(t))#

#cos^2(t)/( cos^2(t)+ sin^2(t))#

Now : # cos^2(t)+ sin^2(t) =1#

RHS is #cos^2(t)#, same as LHS.

QED.

Mar 22, 2018

#"see explanation"#

Explanation:

#"to prove this is an identity either manipulate the left side"#
#"into the form of the right side or manipulate the right side"#
#"into the form of the left side"#

#"using the "color(blue)"trigonometric identities"#

#•color(white)(x)tanx=sinx/cosx" and "sin^2x+cos^2x=1#

#"consider the right side"#

#rArr1/(1+sin^2t/cos^2t)#

#=1/((cos^2t+sin^2t)/cos^2t)#

#=1/(1/cos^2t)#

#=1xxcos^2t/1=cos^2t=" left side hence proved"#