Question

prove identity cot power4 -1=cosec power 4-2 cosec power 2

Answer 1

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${ \cot }^{4} \alpha - 1 = { \csc}^{4} \alpha - 2 { \csc }^{2} \alpha \\ with \: algebric \: identity \\ {a}^{2} - {b}^{2} = (a + b)(a - b) \\ {( \cot }^{2} \alpha - 1)( { \cot }^{2} \alpha + 1) = { \csc }^{2} \alpha ({ \csc}^{2} \alpha - 2) \\ identity \: 1 + {cot}^{2} \alpha = { \csc}^{2} \alpha \\ applying \: we \: get \\ { \csc }^{2} \alpha ( { \cot }^{2} \alpha - 1) = { \csc }^{2} \alpha ({ \csc}^{2} \alpha - 2) \\ ( { \cot }^{2} \alpha - 1) = ({ \csc}^{2} \alpha - 2) \\ 1 = { \csc }^{2} \alpha - { \cot \alpha }^{2} \\ which \: is \: an \: identity$