Mathematics - Application of Derivative Question with Solution | TestHub

$\begin{array}{rl}& \because \tan 65^{\circ }=\tan \left(40^{\circ }+25^{\circ }\right)\\ & \Rightarrow \tan 65^{\circ }=\frac{\tan 40^{\circ }+\tan 25^{\circ }}{1-\tan 25^{\circ }\cdot \tan 40^{\circ }}\\ & \Rightarrow \tan 65^{\circ }-\tan 65^{\circ }\tan 25^{\circ }\tan 40^{\circ }\\ & =\tan 40^{\circ }+\tan 25^{\circ }\\ & \left\{\because \tan 65^{\circ }\tan 25^{\circ }=1\right\}\\ & \Rightarrow \tan 65^{\circ }-\tan 40^{\circ }=\tan 40^{\circ }+\tan 25^{\circ }\end{array}$ Adding $\tan 25^{\circ }$ on both sides, we get $\tan 65^{\circ }+\tan 25^{\circ }=2\left(\tan 40^{\circ }+\tan 25^{\circ }\right)\Rightarrow AP$