We know that tan (A/2) = [sin (A/2)] / [cos (A/2)]
From the half angle formulas of sin and cos,
tan (A/2) = [±√(1 – cos A)/2] / [±√(1 + cos A)/2]
= ±√[(1 – cos A) / (1 + cos A)]
This is one of the formulas of tan (A/2). Let us derive the other two formulas by rationalizing the denominator here.
tan (A/2) = ±√[(1 – cos A) / (1 + cos A)] × √[(1 – cos A) / (1 – cos A)]
= √[(1 – cos A)2 / (1 – cos2A)]
= √[(1 – cos A)2/ sin2A]
= (1 – cos A) / sin A
This is the second formula of tan (A/2). To derive another formula, let us multiply and divide the above formula by (1 + cos A). Then we get
tan (A/2) = [(1 – cos A) / sin A] × [(1 + cos A) / (1 + cos A)]
= (1 – cos2A) / [sin A (1 + cos A)]
= sin2A / [sin A (1 + cos A)]
= sin A / (1 + cos A)
Thus, tan (A/2) = ±√[(1 – cos A) / (1 + cos A)] = (1 – cos A) / sin A = sin A / (1 + cos A).
Leave a Reply