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Prove that the determinant [xsinthetacos...

Prove that the determinant `[xsinthetacostheta-sintheta-x1costheta1x]`is independent of 0.

Text Solution

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`[{:(x,"sin"theta,"cos"theta),("-sin"theta,-x,1),("cos"theta,1,x):}]`
`[{:(-x,1),(1,x):}]"-sin"theta[{:("-sin"theta,1),("cos"theta,x):}]"+cos"theta|{:("-sin"theta,-x),("cos"theta,1):}|`
`=x(-x^(2)-1)-"sin"theta(-x"sin"theta-"cos"theta)+"cos"theta(-"sin"theta"+x"cos"theta)`
`=-x^(3)-x+x"sin"^(2)theta+"sin"theta"cos"theta-"sin"theta"cos"theta+x"cos"^(2)theta`
`=-x^(3)-x+x(sin^(2)theta+cos^(2)theta)`
`=-x^(3)-x+x=-x^(3)`
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Prove that the determinant |[x, sintheta,costheta],[-sintheta,-x,1],[costheta,1,x]| is independent of theta

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Knowledge Check

  • Determinant A=|(x,sintheta,costheta),(-sintheta,-x,1),(costheta,1,x)|

    A
    Independent of `theta`
    B
    dependent of `theta`
    C
    dependent of `theta` and x
    D
    None of the above

  • If sintheta+costheta=x then sintheta-costheta=?

    A
    `pmsqrt(2-x^(2))`
    B
    `pmsqrt(2+x^(2))`
    C
    `pmsqrt(4-x^(2))`
    D
    `pmsqrt(4+x^(2))`

  • (sintheta-costheta+1)/(sintheta+costheta-1) is equal to

    A
    `(1)/(tan theta - sec theta)`
    B
    `(1)/(sec theta - tan theta)`
    C
    `(1)/(cos theta - tan theta)`
    D
    `(1)/(tan theta - cos theta)`

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