Question

prove identities
(sin A +sec A)^2+(cos A+cosec A)^2=(1+sec A cosec A)​

Answer 1

Answer:

(sin A + sec A)² + (Cos A + Cosec A)²

= Sin² A + sec² A + 2 Sin A Sec A + Cos² A + Cosec² A + 2 Cos A Cosec A

(opening the brackets with identity (a+b)²)

= Sin²A + COs²A + Sec²A + Cosec²A + 2 Sin A SecA + 2 Cos A Cosec A

=1  +  [ 1/Cos²A + 1/ Sin²A ] +  [ 2 Sin A / Cos A + 2 Cos A / SIn A ]

(sin²A+cos²A=1) and (sec²A=1/cos²A and cosec²A=1/sin²A)

= 1 + (Sin²A + COs²A)/ [Cos²A Sin²A ]  + 2 [ SIn² A + Cos²A ] / [ SinA CosA

= 1 + 1/Cos²A 1/Sin²A  + 2 1/SinA  1/CosA

= 1 + Sec²A Cosec²A + 2 COsecA Sec A

= (1 + SecA CosecA )²