verify: 1-sin x/1+sin x = (sec x - tan x)^2

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embizze | High School Teacher | (Level 2) Educator Emeritus

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Verify `(1-sinx)/(1+sinx)=(secx-tanx)^2 `

Start with the left hand side:

`(1-sinx)/(1+sinx) `

`=(1-sinx)/(1+sinx) * (1-sinx)/(1-sinx) `

`=(1-sinx)^2/(1-sin^2x) `

`=(1-2sinx+sin^2x)/(1-sin^2x) `

`=(1-2sinx+sin^2x)/(cos^2x) `

`=1/(cos^2x)-(2sinx)/(cos^2x)+(sin^2x)/(cos^2x) `

`=sec^2x-2(sinx)/(cosx)1/(cosx)+tan^2x `

`=sec^2x-2tanxsecx+tan^2x `

`=(secx-tanx)^2 ` as required.

Note that you could start from the right hand side and show that it is equivalent to the left hand side. If you have difficulties, you can rewrite both sides in terms of sinx and cosx and see what happens.