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Questions
cosec4 θ − cosec2 θ = cot4 θ + cot2 θ
Prove the following:
cosec4 θ − cosec2 θ = cot2 θ + cot4 θ
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Solution 1
LHS = cosec4 θ − cosec2 θ
LHS = cosec2 θ (cosec2 θ − 1)
LHS = (cot2 θ + 1)cot2 θ ...`{(cot^2 θ + 1 = cosec^2 θ),(∵ cot^2 θ = cosec^2 θ - 1):}`
LHS = cot4 θ + cot2 θ
RHS = cot4 θ + cot2 θ
RHS = LHS
Hence proved.
Solution 2
RHS = cot4 θ + cot2 θ
RHS = cot2 θ (cot2 θ + 1)
RHS = (cosec2 θ − 1)cosec2 θ ...`{(cot^2 θ + 1=cosec^2 θ),(∵ cot^2θ = cosec^2 θ - 1):}`
RHS = cosec4 θ − cosec2 θ
LHS = cosec4 θ − cosec2 θ
RHS = LHS
Hence proved.
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