Prove that cot2 theta + cosec2 theta = cosec4 theta – cot4 theta. ​

Question

Prove that cot2 theta + cosec2 theta = cosec4 theta – cot4 theta. ​

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Kinsley 3 months 2021-11-06T18:18:03+00:00 1 Answer 0 views 0

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    2021-11-06T18:19:52+00:00

    GIVEN:

    • cot² θ + cosec² θ = cosec⁴ θ – cot⁴ θ

    TO PROVE:

    • cot² θ + cosec² θ = cosec⁴ θ – cot⁴ θ

    FORMUALE USED:

    \boxed{\large{\bold{A^2 - B^2 = (A+B)(A-B)}}}

    \boxed{\large{\bold{cosec^2 \theta - cot^2 \theta = 1 }}}

    PROOF:

    METHOD 1:

    Take cot² θ + cosec² θ as L.H.S and cosec⁴ θ – cot⁴ θ as R.H.S.

    Take the R.H.S

    1(cot² θ + cosec² θ)

    \boxed{\large{\bold{cosec^2 \theta - cot^2 \theta = 1 }}}

    (cosec² θ + cot² θ)(cosec² θ – cot² θ)

    Apply (A + B)(A – B)  = A² – B²

    cosec⁴ θ – cot⁴ θ

    L.H.S =  R.H.S

    cot² θ + cosec² θ = cosec⁴ θ – cot⁴ θ

    HENCE PROVED.

    METHOD 2:

    Take cot² θ + cosec² θ as L.H.S and cosec⁴ θ – cot⁴ θ as R.H.S.

    Take the R.H.S

    cosec⁴ θ – cot⁴ θ

    Apply A² – B² = (A + B)(A – B)

    (cosec² θ + cot² θ)(cosec² θ – cot² θ)

    \boxed{\large{\bold{cosec^2 \theta- cot^2 \theta = 1 }}}

    (cosec² θ – cot² θ) = 1

    (cosec² θ + cot² θ) = L.H.S

    L.H.S = R.H.S

    cot² θ + cosec² θ = cosec⁴ θ – cot⁴ θ

    HENCE PROVED.

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