Question

33. Prove that.
(sin A + cosec A)2 + (cos A + sec A) = 7 + tan- A + cot? A​

Answer 1

(sinA+cosecA)²+(cosA+secA)²

=sin²A+cosec²A+2sinAcosecA+cos²A+sec²A+2cosAsecA

=sin²A+cos²A+cosec²A+sec²A+2sinA×1/sinA+2cosA×1/cosA

=1+cosec²A+sec²A+2+2

=5+(1+cot²A)+(1+tan²A)

=7+tan²A+cot²A

Answer 2

( sinA + 1/sinA )²+ ( cosA + 1/cosA )²

= sin²A + 2 + 1/sin²A + cos²A + 2 + 1/cos²A

As sin²A + cos²A = 1

= 1 + 2 + 2 + cosec²A + sec²

= 5 + ( 1 + cot²A ) + ( 1 + tan²A )

= 5 + 1 + 1 + cot²A + tan²A

= 7 + tan²A + cot²A

Thus proved.

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