Question

Please prove that (1-sin+ cos)(1-sin+ cos) =2(1+cos)(1-sin)

Answer 1

=(1-sin+cos)(1-sin+cos)

=1-sin(1-sin+cos)+cos(1-sin+cos)

=1-sin^2-2sin+cos-cos sin+cos-cos sin+ cos^2

=2+2cos-2sin-2cos sin

=2(1+cos-sin-cos sin)

rhs

2(1-sin+cos-cos sin)

hence lhs=rhs

Answer 2

Answer :-

To Prove:

( 1- sin + cos )(1- sin + cos) = 2 (1+cos)(1- sin)

Proof :

{ (1- sin + cos)(1-sin + cos) can be written as (1- sin +cos)² }

$<font color="</strong><strong>red</strong><strong>">$By using Identity (a-b+c)² = a²+b²+ c² - 2ab - 2bc - 2ac

$<font color="</strong><strong>blac</strong><strong>k</strong><strong>">$Now , By taking L.H.S

= (1 - sin + cos)²

= 1 + sin² + cos ² - 2 sin + 2 cos - 2 sin × cos

= 2 - 2 sin + 2 cos - 2 sin × cos

= 2(1 - sin) + 2 cos (1 - sin)

= 2(1- sin) (1 + cos) = R.H.S

${\fcolorbox{green}{#82ffff}{\textcolor{#300061}{</strong><strong>Hence</strong><strong>,</strong><strong> </strong><strong>proved</strong><strong> }}}$

${\fcolorbox{green}{#82ffff}{\textcolor{#300061}{</strong><strong> </strong><strong>L</strong><strong>.</strong><strong>H</strong><strong>.</strong><strong>S</strong><strong> </strong><strong>=</strong><strong> </strong><strong>R</strong><strong>.</strong><strong>H</strong><strong>.</strong><strong>S</strong><strong> }}}$