Answer:
a) sin x
b) cos² x
c) sin²x
Step-by-step explanation:
Step(i):-
Given that the trigonometric equation
[tex]tan(\frac{x}{2} ) + cos x tan (\frac{x}{2} ) = sin x[/tex]
we have to use trigonometric formulas
[tex]1-cosx = 2sin^{2} (\frac{x}{2} )[/tex]
[tex]sin x = 2 sin \frac{x}{2} cos\frac{x}{2}[/tex]
now
[tex]tan\frac{x}{2} = \frac{sin\frac{x}{2} }{cos\frac{x}{2} }[/tex]
= [tex]\frac{2sin(\frac{x}{2} ) sin\frac{x}{2} )}{2sin\frac{x}{2} cos\frac{x}{2} }[/tex]
= [tex]\frac{2sin^{2} (\frac{x}{2} ) )}{sinx }[/tex]
= [tex]\frac{1-cosx}{sinx}[/tex]
Step(ii):-
[tex]tan(\frac{x}{2} ) + cos x tan (\frac{x}{2} ) = \frac{1-cosx}{sinx} +cosx \frac{1-cosx}{sinx}[/tex]
= [tex]\frac{1-cosx +cosx(1-cosx)}{sinx}[/tex]
= [tex]\frac{1-cosx+cosx-cos^{2} x}{sinx}[/tex]
= [tex]\frac{1-cos^{2} x}{sinx}[/tex]
= [tex]\frac{sin^{2}x }{sinx}[/tex]
= sinx
