Question

tan theta divide by 1- cot theta + cot theta divide by 1-tan theta = 1+ tan theta cot theta

Answer 1

Answer:

$\frac{tan\theta}{1-cot\theta}+\frac{cot\theta}{1-tan\theta}=1+tan\theta+cot\theta$

Step-by-step explanation:

Formula used:

$a^3-b^3=(a-b)(a^2+ab+b^2)$

$\frac{tan\theta}{1-cot\theta}+\frac{cot\theta}{1-tan\theta}$

$=\frac{tan\theta}{1-\frac{1}{tan\theta}}-\frac{cot\theta}{tan\theta-1}$

$=\frac{tan\theta}{\frac{tan\theta-1}{tan\theta}}-\frac{\frac{1}{tan\theta}}{tan\theta-1}$

$=\frac{tan^2\theta}{tan\theta-1}-\frac{\frac{1}{tan\theta}}{tan\theta-1}$

$=\frac{tan^2\theta-\frac{1}{tan\theta}}{tan\theta-1}$

$=\frac{tan^3\theta-1}{tan\theta(tan\theta-1)}$

$=\frac{tan^3\theta-1^3}{tan\theta(tan\theta-1)}$

$=\frac{(tan\theta-1)(tan^2\theta+tan\theta+1)}{tan\theta(tan\theta-1)}$

$=\frac{tan^2\theta+tan\theta+1}{tan\theta}$

$=\frac{tan^2\theta}{tan\theta}+\frac{tan\theta}{tan\theta}+\frac{1}{tan\theta}$

$=tan\theta+1+cot\theta$

$=1+tan\theta+cot\theta$