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secondary 3 | A Maths

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how to prove the following identity pls help thank you

(Property applied here is a² - b² = (a + b)(a - b) )

② Change 1 + 2sinxcosx to sin²x + cos²x + 2sinxcosx, (use property cos²x + sin²x = 1)

which = (sinx + cosx)²

(Property used here is (a + b)² = a² + 2ab + b²)

③ Then cancel out (cosx + sinx) on both the numerator and denominator. This leaves (cosx + sinx)/(cosx - sinx)

④ Divide both numerator and numerator by cosx.

This leaves (1 + sinx/cosx)/(1 - sinx/cosx)

, which equals (1 + tanx)/(1 - tanx)

(1 + 2sinxcosx)/(cos²x - sin²x)

= (sin²x + cos²x + 2sinxcosx)/(cos²x - sin²x)

= (sin²x/cos²x + cos²x/cos²x + 2sinxcosx/cos²x)/(cos²x/cos²x - sin²x/cos²x)

= (tan²x + 2sinx/cosx + 1)/(1 - tan²x)

= (tan²x + 2 tanx + 1)/( (1 + tanx)(1 - tanx) )

= (tanx + 1)² / ( (1 + tanx)(1 - tanx) )

= (tanx + 1)/(1 - tanx)

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