How do you rewrite the expression as a single logarithm and simplify #ln(cos^2t)+ln(1+tan^2t)#?

2 Answers
Alan P.
Jan 13, 2017

#ln(cos^2(t))+ln(1+tan^2(t)) = color(green)0#

Explanation:

Things to remember:

[1]#color(white)("XXX")color(red)(tan=sin/cos)#

[2]#color(white)("XXX")color(red)(ln(a)+ln(b)=ln(a * b)#

[3]#color(white)("XXX")color(red)(cos^2+sin^2=1#

[4]#color(white)("XXX")color(red)(ln(a)=k if e^k=a)#

Therefore
#ln(cos^2(t))+ln(1+tan^2(t))#

#color(white)("XXX")=ln(cos^2(t)+sin^2(t))#

#color(white)("XXX")=ln(1)#

#color(white)("XXX")=0#

Ratnaker Mehta
Jan 13, 2017

#0#.

Explanation:

We have to use the Rule # : ln a+ln b=ln(ab)#, together with the

Identity # : 1+tan^2t=sec^2t#

The Exp. #=lncos^2t+ln(1+tan^2t)#

#=lncos^2t+lnsec^2t#

#=ln{(cos^2t)(sec^2t)}#

#=ln 1#

#=0#.

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