Maths

Prove sqrt(Sec^2 A+Cosec^2 A)=TanA+CotA

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  1. It is not true if (for example) A=3pi/4

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  2. Expand left side into sines and cosines:
    sqrt(sec^2A+csc^2A)
    =sqrt(1/cos^2A+1/sin^2A)
    =sqrt((sin^2A+cos^2A)/(cos^2A sin^2A))
    =sqrt(1/(cos^2A sin^2A))
    =1/cosA sinA

    Similarly expand right hand side:
    tanA+cotA
    =sinA/cosA + cosA/sinA
    =(sin^2A + cos^2A)/(cosA sinA)
    =1/(cosA sinA)

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  3. tan(3pi/4)=cot(3pi/4)=-1
    sec^2(3pi/4)=csc^2(3pi/4)=2
    sqrt(2+2)=-1-1 ?

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  4. Here square-root is taken of the square of the product of two functions, and not the numerical values.

    To me it is justified to retain the signs of the original functions, namely sin(x) and cos(x) in the square-root.

    So if we evaluate the functions after taking square-root,
    LHS=1/(cos(3π/4)sin(3π/3)=-2
    and
    RHS=-2 as you have calculated.

    As a compromise, we can say that the identity should read:
    (Sec^2 A+Cosec^2 A)=(TanA+CotA)²

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  5. In the problem we must add
    0<A<pi/2 that's all

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  6. Yes, or in more general terms
    kπ<A<kπ+π/2 k∈Z

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