How do you verify $(1+sinx)/(1-sinx) - (1-sinx)/(1+sinx)=4tanxsecx$ ?

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Answer 1 · 2015-11-01T04:47:37

See explanation.

$[1]" "(1+sinx)/(1-sinx)-(1-sinx)/(1+sinx)$

Combine the two terms by making them have the same denominator.

$[2]" "=((1+sinx)/(1-sinx))((1+sinx)/(1+sinx))-((1-sinx)/(1+sinx))((1-sinx)/(1-sinx))$

$[3]" "=(1+2sinx+sin^2x)/(1-sin^2x)-(1-2sinx+sin^2x)/(1-sin^2x)$

$[4]" "=(1+2sinx+sin^2x-1+2sinx-sin^2x)/(1-sin^2x)$

$[5]" "=(4sinx)/(1-sin^2x)$

Pythagorean Identity: $1-sin^2theta=cos^2theta$

$[6]" "=(4sinx)/(cos^2x)$

$[7]" "=(4sinx)/((cosx)(cosx))$

Quotient Identity: $sintheta/costheta=tantheta$

$[8]" "=(4tanx)/(cosx)$

Reciprocal Identity: $1/costheta=sectheta$

$[9]" "=4tanxsecx$

$color(blue)("":.(1+sinx)/(1-sinx)-(1-sinx)/(1+sinx)=4tanxsecx)$

How do you verify (1+sinx)/(1-sinx) - (1-sinx)/(1+sinx)=4tanxsecx (1+sinx)/(1-sinx) - (1-sinx)/(1+sinx)=4tanxsecx?