(Q19)Prove that (sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A

L.H.S. = (sin A + cosec A)2 + (cos A + sec A)2

= (2 A + 2 A + sin A . cosec A) + (2 A – 2 A + cos A . sec A)

[∵ (a + b)2 = a2 + b2 + 2ab]

= (2 A + 2 A) + cosec2 A + sin A .
1
A
 + sec2 A + cos A .
1
A
[∵
1
sin A
 = cosec A;
1
cos A
 = sec A]

= +(1 + 2 A) + 2 + (1 + 2 A) + 2

[∵ sin2 A + cos2 A = 1; cosec2 A = 1 + cot2 A; sec2 A = 1 + tan2 A]

= + 2 A + 2 A

= R.H.S.