Prove that (\sin A + \cosec A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A

Step 1: Use the (a + b)^2 formula to simplify the given equation.
EXAMPLE: L.H.S
= (\sin A + \cosec A)^2 + (\cos A + \sec A)^2 (Given)
= (\sin^2 A + \cosec^2 A + 2 \sin A \cosec A) + (\cos^2 A + \sec^2 A + 2 \cos A sec A) (Since, (a + b)^2 = a^2 + b^2 + 2ab)
Step 2: Use the trigonometric ratios to simplify
EXAMPLE: \sin^2 A + \cosec^2 A + 2 \sin A \frac{1}{\sin A} + \cos^2 A + \sec^2 A + 2 \cos A \frac{1}{cos A}
= \sin^2 A + \cos^2 A + \cosec^2 A + \sec^2 A + 2 + 2
= 1 + (1 + \cot^2 A ) + (1 + \tan^2 A) + 4
(Since, \sin^2 A + \cos^2 A = 1
1 + \cot^2 A = \cosec^2 A
1 + \tan^2 A = \sec^2 A )
= 1 + 1 + \cot^2 A + 1 + \tan^2 A + 4
= 7 + \cot^2 A + \tan^2 A
R.H.S proved