Question

Find value of $Q$ -

$\frac{1}{\sec Q}+\frac{1}{\sec Q+1}=2\csc Q\times \cot Q$

Answer 1

$\frac{1}{secQ}+\frac{1}{secQ+1}=2cosecQ\times cotQ$

$cosQ+\frac{1}{secQ+1}=\frac{2}{\sin Q}\times \frac{cosQ}{\sin Q}$

$\frac{\cos Q\sec Q+\cos Q+1}{secQ+1}=\frac{2cosQ}{{\sin }^{2}Q}$

$\frac{2+\cos Q}{secQ+1}=\frac{2cosQ}{{\sin }^{2}Q}$

$\left(2+\cos Q\right)\left({\sin }^{2}Q\right)=\left(2cosQ\right)\left(secQ+1\right)$

$\left(1-{\cos }^{2}Q\right)\left(2+\cos Q\right)=2+2cosQ$

$\left(1-\cos Q\right)\left(1+\cos Q\right)\left(2+\cos Q\right)=2\left(1+cosQ\right)$

$\left(1+\cos Q\right)\left[\left(1-\cos Q\right)\left(2+\cos Q\right)-2\right]=0$

$\left(1+\cos Q\right)\left[2+\cos Q-2\cos Q-{\cos }^{2}Q-2\right]=0$

$\left(1+\cos Q\right)\left[-\cos Q-{\cos }^{2}Q\right]=0$

$\left(1+\cos Q\right)\cos Q\left(1+\cos Q\right)=0$

$\cos Q=0$ or $1+\cos Q=0$

$\cos Q=0$ or $\cos Q=-1$

$Q=\frac{\pi }{2},\pi$