How do you prove 1cos(x)sin(x)=tan(x2)?

2 Answers
Oct 1, 2016

Please see below.

Explanation:

As cosx=12sin2(x2) and sinx=2sin(x2)cos(x2)

Hence 1cosxsinx

= 1(12sin2(x2))2sin(x2)cos(x2)

= 11+2sin2(x2)2sin(x2)cos(x2)

= 2sin2(x2)2sin(x2)cos(x2)

= 2sin(x2)sin(x2)2sin(x2)cos(x2)

= sin(x2)cos(x2)

= tan(x2)

Oct 1, 2016

Please follow the instructions below:

Explanation:

sin(x2+x2)

=sin(x2)cos(x2)+cos(x2)sin(x2)

=sin(x2){cos(x2)+cos(x2)}

=sin(x2)2cos(x2)

=2sin(x2)cos(x2)

=sin(x)

This is because:

sin(A+B)=sin(A)cos(B)+cos(A)sin(B)

cos(x2+x2)

=cos(x2)cos(x2)sin(x2)sin(x2)

=cos2(x2)sin2(x2)

=1sin2(x2)sin2(x2)

=12sin2(x2)

=cos(x)

This is because:

cos(A+B)=cos(A)cos(B)sin(A)sin(B)

Which means that:

LHS

=1cos(x)sin(x)

=1{12sin2(x2)}2sin(x2)cos(x2)

=11+2sin2(x2)2sin(x2)cos(x2)

=2sin2(x2)2sin(x2)cos(x2)

=2sin(x2)sin(x2)2sin(x2)cos(x2)

=sin(x2)cos(x2)

=tan(x2)

=RHS