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

Answers ( )

    0
    2021-10-10T13:29:27+00:00

    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!

    0
    2021-10-10T13:29:45+00:00

    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^

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