The identity `(tanx-sinx)/(tanx+sinx)=(secx-1)/(secx+1)` has to be verified

`(tanx-sinx)/(tanx+sinx)`

=> `((sin x/cos x)-sinx)/((sin x/cos x)+sinx)`

=> `(sin x(1/cos x-1))/(sin x(1/cos x+1))`

=> `(1/cos x-1)/(1/cos x+1)`

=> `(sec x - 1)/(sec x + 1)`

**This proves that **`(tanx-sinx)/(tanx+sinx)=(secx-1)/(secx+1)`

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The identity `(tanx-sinx)/(tanx+sinx)=(secx-1)/(secx+1)` has to be verified

`(tanx-sinx)/(tanx+sinx)`

=> `((sin x/cos x)-sinx)/((sin x/cos x)+sinx)`

=> `(sin x(1/cos x-1))/(sin x(1/cos x+1))`

=> `(1/cos x-1)/(1/cos x+1)`

=> `(sec x - 1)/(sec x + 1)`

**This proves that **`(tanx-sinx)/(tanx+sinx)=(secx-1)/(secx+1)`