Question 1183638
Prove each of the following trigonometric identities.

1) 


{{{sin (x) sin (2x) + cos (x )cos (2x) = cos( x)}}}


manipulate left side


{{{sin (x) *sin (2x) + cos (x )*cos (2x)}}} ..........use {{{sin (2x) =2sin(x)* cos(x) and cos (2x)=cos^2(x) - sin^2(x)}}}


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


= {{{2sin^2(x)* cos(x)+cos (x )* (cos^2(x) - sin^2(x))}}}...factor out {{{cos(x)}}}


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


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


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


= {{{cos(x)}}}



2) 


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


manipulate right side


{{{sin (x )sin (pi/2 - x) + cos^2( x) cot( x)}}}.........use {{{sin(pi/2-x)=sin(pi/2)*cos(x)-cos(pi/2)*sin(x)= cos(x)=1*cos(x) - 0*sin(x)=cos(x) }}}


={{{sin (x )cos(x) + cos^2( x) cot( x)}}}.... use {{{cot( x)=cos(x)/sin (x )}}}


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


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


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


={{{(1 *cos(x))/sin (x ))}}}


={{{cos(x)/sin (x )}}}


={{{cot( x)}}}



3) 


{{{2csc (2x) = sec (x) *csc( x)}}}


manipulate left side


{{{2csc (2x)}}} ..................use {{{csc (2x) = (1/2 )csc(x) sec(x)}}}


={{{2(1/2)csc(x) *sec(x))}}} .....simplify


={{{csc(x) *sec(x))}}}