Question

C= sin^2a - tan^2a / cos^2a - cot^2a

Answer 1

\(C=\dfrac{sin^2a-tan^2a}{cos^2a-cot^2a}\)

\(=\dfrac{sin^2a-\dfrac{sin^2a}{cos^2a}}{cos^2a-\dfrac{cos^2a}{sin^2a}}\)

\(=\dfrac{sin^2a\left(1-\dfrac{1}{cos^2a}\right)}{cos^2a\left(1-\dfrac{1}{sin^2a}\right)}=tan^2a\cdot\dfrac{\dfrac{cos^2a-1}{cos^2a}}{\dfrac{sin^2a-1}{sin^2a}}\)

\(=tan^2a\cdot\left(\dfrac{cos^2a-1}{cos^2a}\cdot\dfrac{sin^2a}{sin^2a-1}\right)\)

\(=tan^2a\left(\dfrac{1-cos^2a}{1-sin^2a}\cdot tan^2a\right)\)

\(=tan^2a\cdot\left(\dfrac{sin^2a}{cos^2a}\cdot tan^2a\right)=tan^2a\cdot\left(tan^2a\cdot tan^2a\right)\)

\(=tan^6a\)