Simplify $\frac{cos^{2}a - cot^{2} a+1}{sin^{2}a+tan^{2}a -1}$

Question

solomonzelman · Answer

I would start from simplifying the denominator

$\Large\color{orangered}{   sin^2a  +tan^2a+1 }$

$\Large\color{orangered}{   sin^2a  +\frac{sin^2a}{cos^2a}+1 }$

$\Large\color{orangered}{   \frac{sin^2a~cos^2a}{cos^2a}  +\frac{sin^2a}{cos^2a}+\frac{cos^2a}{cos^2a} }$

$\Large\color{orangered}{    \frac{sin^2a~cos^2a+ sin^2a+ cos^2a}{cos^2a} }$

then the numerator,

$\Large\color{orangered}{   cos^2a  +cot^2a+1 }$

$\Large\color{orangered}{   cos^2a  +\frac{cos^2a}{sin^2a}+1 }$

$\Large\color{orangered}{  \frac{cos^2a~sin^2a}{sin^2a} +\frac{cos^2a}{sin^2a}+\frac{sin^2a}{sin^2a} }$

$\Large\color{orangered}{    \frac{sin^2a~cos^2a+ sin^2a+ cos^2a}{cos^2a} }$

solomonzelman · Answer

yeah, the denominator and numerator are equal. I switched the order a little but a+b=b+a so it doesn't matter. The answer is 1.

anonymous · Answer

Thank you @SolomonZelman :D

solomonzelman · Answer

Anytime