The value of (sin A + cosec A)² + (cos A + sec A)² is:
7 + tan2A + cot2A
Expand each square using \((a+b)^2 = a^2 + 2ab + b^2\).
First term: \((\sin A + \csc A)^2 = \sin^2 A + 2\sin A \cdot \csc A + \csc^2 A = \sin^2 A + 2 + \csc^2 A\), since \(\sin A \cdot \csc A = 1\).
Second term: \((\cos A + \sec A)^2 = \cos^2 A + 2 + \sec^2 A\).
Add them: \((\sin^2 A + \cos^2 A) + 4 + \csc^2 A + \sec^2 A = 1 + 4 + (1 + \cot^2 A) + (1 + \tan^2 A)\), using \(\csc^2 A = 1 + \cot^2 A\) and \(\sec^2 A = 1 + \tan^2 A\).
This simplifies to \(7 + \tan^2 A + \cot^2 A\).
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