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Prove: sec^4A(1-sin^4A)-2tan^2A=1...

Prove: `sec^4A(1-sin^4A)-2tan^2A=1`

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Proof:
`LHS=sec^(4)A(1-sin^(4)A)-2tan^(2)A`
`=sec^(4)A-sec^(4)A.sin^(4)A-2tan^(2)A`
`=1/(cos^(4)A)-1/(cos^(4)A).sin^(4)A-2xx(sin^(2)A)/(cos^(2)A)`
`…….(secA=1/(cosA),tanA=(sinA)/(cosA))`
`=1/(cos^(4)A)-(sin^(4)A)/(cos^(4)A)-(2sin^(2)A)/(cos^(2)A)`
`=((1-sin^(4)A))/(cos^(4)A)-(2sin^(2)A)/(cos^(2)A)`
`=([1-(sin^(2)A)^(2)])/(cos^(4)A)-(2sin^(2)A)/(cos^(2)A)`
`=((1+sin^(2)A)(1-sin^(2)A))/(cos^(4)A)-(2sin^(2)A)/(cos^(2)A)`
`..........[a^(2)-b^(2)=(a+b)(a-b)]`
`=((1+sin^(2)A)xxcos^(2)A)/(cos^(4)A)-(2sin^(2)A)/(cos^(2)A)...........[sin^(2)A+cos^(2)A=1]`
`=(1+sin^(2)A)/(cos^(2)A)-(2 sin^(2)A)/(cos^(2)A)`
`=(1+sin^(2)A-2sin^(2)A)/(cos^(2)A)`
`=(1-sin^(2)A)/(cos^(2)A)`
`(cos^(2)A)/(cos^(2)A)`......`[sin^(2)A+cos^(2)A=1]`
`=1=RHS`
`:.sec^(4)A(1-sin^(4)A)-2tan^(2)A=1`.
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