if sin@ + cosec@ =√3 prove that Sin^2@+cosec^2@=1​

Question

if sin@ + cosec@ =√3 prove that Sin^2@+cosec^2@=1​

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Ruby 4 weeks 2021-09-17T08:37:38+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-09-17T08:39:25+00:00

    Answer:

    SinA+cosecA=v3 is impossible

    SinA+cosec A greater than or equal to 2

    0
    2021-09-17T08:39:31+00:00

    Given :

    • sin A + cosec A = √3

    To prove :

    • sin²A + cosec²A = 1

    Proof :

    As given

    → sin A + cosec A = √3

    squaring both sides

    → ( sin A + cosec A )² = ( √3 )²

    using algebraic identity

    ( a + b )² = a² + b² + 2 a b

    → sin²A + cosec²A + 2 sin A cosec A = 3

    putting cosec A = 1 / sin A

    → sin²A + cosec²A + 2 sin A ( 1 / sin A ) = 3

    → sin²A + cosec²A + 2 = 3

    → sin²A + cosec²A = 3 – 2

    sin²A + cosec²A = 1

    Proved .

    Trigonometric ratios and identities to know :

    __________________________

    →  Cos² θ + Sin² θ = 1

    → 1 + Tan² θ = Sec² θ

    → 1 + Cot² θ = Cosec² θ

    __________________________

    →  sin ( 90 – A ) = cos A

    → cos ( 90 – A ) = sin A

    → tan ( 90 – A ) = cot A

    → cot ( 90 – A ) = tan A

    →  sec ( 90 – A ) = cosec A

    → cosec ( 90 – A ) = sec A

    __________________________

    →  cosec A = 1 / sin A

    →  sec A = 1 / cos A

    →  tan A = 1  / cot A = sin A / cos A

    → cot A = 1 / tan A = cos A / sin A

    __________________________

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