Question

Solve
$(\sin A+\csc A{)}^2+(\cos A+\sec A{)}^2=7+{\tan }^{2}A+{\cot }^{2}A$

Answer 1

We have

${\left(\sin A+\cos ecA\right)}^2+{\left(\cos A+\sec A\right)}^2$

$={\sin }^{2}A+\cos e{c}^{2}A+2.\sin A.\cos ecA+{\cos }^{2}A+{\sec }^{2}A+2\cos A.\sec A$

$=1+\cos e{c}^{2}A+2+{\sec }^{2}A+2\left[\therefore {\sin }^2\theta +{\cos }^2\theta =1\right]$

$=1+1+{\cot }^{2}A+2+1+{\tan }^{2}A+2\left[\cos e{c}^2\theta =1+{\cot }^2\theta \&{\sec }^2\theta =1+{\tan }^2\theta \right]$

$=7+{\cot }^{2}A+{\tan }^{2}A$

Hence

${\left(\sin A+\cos ecA\right)}^2+{\left(\cos A+\sec A\right)}^2=7+{\cot }^{2}A+{\tan }^{2}A$