Question

Substitute x = 9tanθ into the expression $\sqrt{81+ x^{2} } /x$

Answer 1

$\dfrac{\sqrt{81+x^2}}x=\dfrac{\sqrt{81+(9\tan\theta)^2}}{9\tan\theta}=\dfrac{\sqrt{81+81\tan^2\theta}}{9\tan\theta}=\dfrac{9\sqrt{1+\tan^2\theta}}{9\tan\theta}=\dfrac{\sqrt{\sec^2\theta}}{\tan\theta}=\dfrac{|\sec\theta|}{\tan\theta}$