How do you prove the identity $[1-sinx]/secx = (cos^3x)/(1+sinx)$ ?

Question

$[1-sinx]/secx = (cos^3x)/(1+sinx)$

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Answer 1 · 2018-04-07T10:42:31

After editing the question:
$(1-sinx)/secx =color(red) (cos3x)/(1+sinx)to (1-sinx)/secx = color(red)(cos^3x)/(1+sinx)$

We know that,

$color(green)((1)sin^2theta+cos^2theta=1=>cos^2theta=1- sin^2theta$

$color(green)((2)sectheta=1/costheta$

We have to prove,

$(1-sinx)/secx = (cos^3x)/(1+sinx)$

We take

$LHS=(1-sinx)/secx$

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

$=1/(secx)xx(1-sin^2x)/(1+sinx)...toApply (1) and (2)$

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

$=cos^3x/(1+sinx)$

$=RHS$