## Prove that:- (sinA+cosecA) ²+(cosA+secA) ²=7+tan²A+cot²A​

Question

Prove that:-
(sinA+cosecA) ²+(cosA+secA) ²=7+tan²A+cot²A​

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2 months 2021-10-10T13:27:57+00:00 2 Answers 0 views 0

1. ## • To Prove :

(sinA + cosecA)² + (cosA + secA)² = 7 + tan²A + cot²A

## • Proof :

(sinA + cosecA)² + (cosA + secA)²

(a + b)² = a² + b² + 2ab

( sin²A + cosec²A + 2 × sinA × cosecA ) + ( cos²A + sec²A + 2 × cosA × secA )

cosecA = 1 / sinA

secA = 1 / cosA

⇒ ( sin²A + cosec²A + 2 × sinA × 1 /sinA ) + ( cos²A + sec²A + 2 × cosA × 1 /cosA )

⇒ sin²A + cosec²A + 2 + cos²A + sec²A + 2

⇒ ( sin²A + cos²A ) + cosec²A + sec²A + 4

( sin²A + cos²A ) = 1

⇒ 1 + cosec²A + sec²A + 4

⇒ 5 + cosec²A + sec²A

cosec²A = 1 + cot²A

sec²A = 1 + tan²A

⇒ 5 + 1 + cot²A + 1 + tan²A

7 + tan²A + cot²A Hence, Proved!

2. Step-by-step explanation:

=>(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

Identities used:

1+tan²A=sec²A

1+cot²A=cosec²A

sin²A+cos²A=1

cosecA=1/sinA

secA=1/cosA

_________hence Proved …..

_____√\/\______Anushka ^o^