Question

Prove that :$(\sin A+\sec A{)}^2+(\cos A+cosecA{)}^2=(1+\sec AcosecA{)}^2$.

Answer 1

$LHS={(\sin A+\sec A)}^2+{(\cos A+cosecA)}^2={\sin }^{2}A+2\sin A\sec A+{\sec }^{2}A+{\cos }^{2}A+2\cos AcosecA+{cosec}^{2}A=({\sin }^{2}A+{\cos }^{2}A)+({\sec }^{2}A+{cosec}^{2}A)+2\sin A\sec A+2\cos AcosecA=1+(\frac{1}{{\cos }^{2}A}+\frac{1}{{\sin }^{2}A})+2\frac{\sin A}{\cos A}+2\frac{\cos A}{\sin A}(\because {\sin }^{2}A+{\cos }^{2}A=1)=1+\left(\frac{{\sin }^{2}A+{\cos }^{2}A}{{\cos }^{2}A{\sin }^{2}A}\right)+2\left(\frac{{\sin }^{2}A+{\cos }^{2}A}{\cos A\sin A}\right)=1+\left(\frac{1}{{\cos }^{2}A{\sin }^{2}A}\right)+2\left(\frac{1}{\cos A\sin A}\right)=1+{\sec }^{2}A{cosec}^{2}A+2\sec AcosecA={(1+\sec AcosecA)}^2=RHS$