Prove that $(1+sinx)/(1-sinx)=(secx+tanx)^2$ ?

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Answer 1 · 2016-10-13T10:27:28

Using the identities

we have

$(1+sin(x))/(1-sin(x)) = ((1+sin(x))(1+sin(x)))/((1-sin(x))(1+sin(x)))$

$=(1+sin(x))^2/(1-sin^2(x))$

$=(1+sin(x))^2/cos^2(x)$

$=((1+sin(x))/cos(x))^2$

$=(1/cos(x)+sin(x)/cos(x))^2$

$=(sec(x)+tan(x))^2$

Answer 2 · 2016-10-13T10:27:46

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

Let us start from right hand side.

$(secx+tanx)^2$

= $(1/cosx+sinx/cosx)^2$

= $((1+sinx)/cosx)^2$

= $(1+sinx)^2/cos^2x$

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

= $(1+sinx)^2/(1+sinx)(1-sinx)$

= $(1+sinx)/(1-sinx)$

Answer 3 · 2016-10-13T10:31:22

$LHS=(1+sinx)/(1-sinx)$

$=((1+sinx)(1+sinx))/((1-sinx)(1+sinx))$

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

$=(1+sinx)^2/cos^2x$

$=((1+sinx)/cosx)^2$

$=(1/cosx+sinx/cosx)^2$

$=(secx+tanx)^2$

Proved

Prove that (1+sinx)/(1-sinx)=(secx+tanx)^2(1+sinx)/(1-sinx)=(secx+tanx)^2?