Question 930153

1.

{{{(sec( x)-tan( x))(sin (x)+1)= cos (x)}}} 

start with


{{{(sec( x)-tan( x))(sin (x)+1)}}}..............use identities {{{sec( x)=1/cos(x)}}} and {{{tan( x)(sin( x)/cos(x))}}}


={{{(1/cos(x) -sin( x)/cos(x))(sin (x)+1)}}}


={{{(1/cos(x) -sin( x)/cos(x))(sin (x)+1)}}}


={{{((1-sin( x))(sin (x)+1))/cos(x)}}}


={{{(1^2-sin^2( x))/cos(x)}}}


={{{(1-sin^2( x))/cos(x)}}} .......use identity  {{{(1-sin^2( x))=cos^2(x)}}} 


={{{cos^2(x) /cos(x)}}}


={{{cos^cross(2)(x) /cross(cos(x))}}}


={{{cos(x) }}}




2.

{{{1-sin^2(x) = cos(x)/ (tan(x)+cot(x))}}}...I am not quite sure what you need here, this is not true statement



3.
{{{(tan^2 (x )- 1)/( tan^2( x) + 1) = 1 - 2 cos^2( x)}}}

start with

{{{(tan^2 (x)-1)/(tan^2(x)+ 1)}}}


={{{(sin^2(x)/cos^2(x) -1)/( sin^2(x)/cos^2(x) + 1) }}}


={{{(sin^2 (x)/cos^2(x)-1*cos^2(x)/cos^2(x))/( sin^2 (x)/cos^2(x) + 1*cos^2(x)/cos^2(x)) }}}


={{{((sin^2 (x) -cos^2(x))cos^2(x))/( (sin^2 (x) + cos^2(x))/cos^2(x)) }}}


={{{(cross(cos^2(x))(sin^2 (x) - cos^2(x)))/( (sin^2 (x)+ cos^2(x)) *cross(cos^2(x)))}}}


 ={{{((sin^2(x) - cos^2(x)))/( (sin^2 (x)+ cos^2(x)))}}}...{{{sin^2(x)=1-cos^2(x)}}}


={{{((1-cos^2(x) - cos^2(x)))/( (1-cross(cos^2 (x))+cross( cos^2(x))))}}}


={{{((1-2cos^2(x))/1)}}}


={{{1-2cos^2(x)}}}