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Mathematics

If A = 30°; show that :

1 - cos 2 Asin 2 A\dfrac{\text{1 - cos 2 A}}{\text{sin 2 A}} = tan A

Trigonometric Identities

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Answer

1 - cos 2 Asin 2 A\dfrac{\text{1 - cos 2 A}}{\text{sin 2 A}} = tan A

L.H.S.=1 - cos 2 Asin 2 A=1 - cos (2 x 30°)sin (2 x 30°)=1 - cos 60°sin 60°=1−1232=22−1232=2−1232=1232=1232=13\text{L.H.S.} = \dfrac{\text{1 - cos 2 A}}{\text{sin 2 A}}\\[1em] = \dfrac{\text{1 - cos (2 x 30°)}}{\text{sin (2 x 30°)}}\\[1em] = \dfrac{\text{1 - cos 60°}}{\text{sin 60°}}\\[1em] = \dfrac{1 - \dfrac{1}{2}}{\dfrac{\sqrt3}{2}}\\[1em] = \dfrac{\dfrac{2}{2} - \dfrac{1}{2}}{\dfrac{\sqrt3}{2}}\\[1em] = \dfrac{\dfrac{2 - 1}{2}}{\dfrac{\sqrt3}{2}}\\[1em] = \dfrac{\dfrac{1}{2}}{\dfrac{\sqrt3}{2}}\\[1em] = \dfrac{\dfrac{1}{\cancel2}}{\dfrac{\sqrt3}{\cancel2}}\\[1em] = \dfrac{1}{\sqrt3}\\[1em]

R.H.S. = tan A

= tan 30°

= 13\dfrac{1}{\sqrt3}

∴ L.H.S. = R.H.S.

Hence, 1 - cos 2 Asin 2 A\dfrac{\text{1 - cos 2 A}}{\text{sin 2 A}} = tan A

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