Question

Question #58780

Answer 1

`RHS=(sin^2x)/(2+sin2xcscx)`

`=(2sin(x/2)cos(x/2))^2/(2+2sinxcosx xx1/sinx)`

`=(4sin^2(x/2)cos^2(x/2))/(2(1+cosx ))`

`=(4sin^2(x/2)cos^2(x/2))/(2*2cos^2(x /2))`

`=sin^2(x/2)=LHS`

Answer 2

Formulas to be used in the proof.

1. `sin 2x = 2 sin x cos x`

2. `sin (1/2 x)=+-sqrt((1-cosx)/2)`
   `:.sin ^2(1/2 x)=(1-cosx)/2`

3. `cscx=1/sinx`
4. `sin^2x+cos^2x=1`
   `sin^2x=1-cos^2x`

Right Hand Side:

`sin^2x/(2+sin2xcscx)=sin^2x/(2+2sinx cosx cscx)`

`=sin^2x/(2+2cancelsinx cosx * 1/cancelsinx)`

`=(1-cos^2x)/(2+2cosx )`

`=((1-cosx)(1+cosx))/(2(1+cosx ))`

`=((1-cosx)cancel(1+cosx))/(2cancel(1+cosx ))`

`=(1-cosx)/2`

`=sin^2(1/2)x`

`:.=`Left Hand Side