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)#