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    If sin⁡θ−cosecθ=√2sin⁡θ-cosec θ=√2sin⁡θ−cosecθ=√2​, then the value of  sin3⁡θ−cosec3θsin^3⁡θ-cosec^3 θsin3⁡θ−cosec3θ​ is:  
    Question

    If sinθcosecθ=2sin⁡θ-cosec θ=√2​, then the value of  sin3θcosec3θsin^3⁡θ-cosec^3 θ​ is:  

    A.

    2√3

    B.

    0

    C.

    1/√2

    D.

    5√2

    Correct option is D

    Given: 
    sinθcosecθ=2sin⁡θ-cosec θ=√2​, 
    To find ;  sin3θcosec3θsin^3⁡θ-cosec^3 θ 
    Formula Used: 
    As we know that if x1x=kx - \frac{1}{x} = k 

    than, x31x3=K3+3Kx^3 - \frac{1}{x^3} = K^3 + 3K ​

    sinθ=1cosecθ\sin \theta = \frac{1}{\cosec \theta}   

    Solution:  
    sinθcosecθ=2\sin \theta - \cosec \theta = \sqrt{2}   
    sinθ1sinθ=2\sin \theta - \frac{1}{\sin \theta} = \sqrt{2}    
    let sinθ=x\sin \theta = x than
    Using the given formula 
    sin3θcosec3θ=(2)3+3×2                      =22+32                      =52\sin^3 \theta - \cosec^3 \theta =( \sqrt{2})^3 + 3 \times \sqrt2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =2\sqrt 2+3\sqrt2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 5 \sqrt 2​​

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