Question

Find the value of $(\sin \theta +\sec \theta {)}^2+(\cos \theta +\text{cosec}\theta {)}^2-(1+\sec \theta \text{cosec}\theta {)}^2$

Answer 1

${(\cos \theta +\csc \theta )}^2+{(\sin \theta +\sec \theta )}^2-(1+\sec \theta \csc \theta {)}^2$

$⟹{\cos }^2\theta +{\csc }^2\theta +2\cos \theta \csc \theta +{\sin }^2\theta +2\sin \theta \sec \theta +{\sec }^2\theta -(1+2\sec \theta \csc \theta +{\csc }^2\theta )$

$⟹({\sin }^2\theta +{\cos }^2\theta )+{\csc }^2\theta +{\sec }^2\theta -2\left(\frac{\cos \theta }{\sin \theta }\right)+2\left(\frac{\sin \theta }{\cos \theta }\right)-1-{\sec }^2\theta {\csc }^2\theta -2\sec \theta \csc \theta$

$⟹1+\frac{1}{{\sin }^2\theta }+\frac{1}{{\cos }^2\theta }+2\left(\frac{{\cos }^2\theta +{\sin }^2\theta }{\sin \theta \cos \theta }\right)-1-2\sec \theta \csc \theta -{\sec }^2\theta {\csc }^2\theta$

$⟹\left(\frac{{\cos }^2\theta +{\sin }^2\theta }{{\sin }^2\theta {\cos }^2\theta }\right)+2\left(\frac{1}{\sin \theta \cos \theta }\right)-2\sec \theta \csc \theta -{\sec }^2\theta {\csc }^2\theta$

$⟹\left(\frac{1}{{\sin }^2\theta {\cos }^2\theta }\right)+2\sec \theta \csc \theta -2\sec \theta \csc \theta -{\sec }^2\theta {\csc }^2\theta$

$⟹{\sec }^2\theta {\csc }^2\theta -{\sec }^2\theta {\csc }^2\theta$

$⟹0$