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Mathematics

Verify each of the following :

(i) sin 60° cos 30° - cos 60° sin 30° = sin 30°

(ii) 2 sin 30° cos 30° = sin 60°

(iii) 2 sin 45° cos 45° = sin 90°

Trigonometrical Ratios

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Answer

(i) Left Hand Side :

sin 60° cos 30° - cos 60° sin 30°

= 32×32−12×12\dfrac{\sqrt{3}}{2}\times \dfrac{\sqrt{3}}{2} -\dfrac{1}{2}\times \dfrac{1}{2}

= 34−14=24\dfrac{3}{4} -\dfrac{1}{4} = \dfrac{2}{4}

= 12\dfrac{1}{2}.

Right Hand Side :

sin 30° = 12\dfrac{1}{2}

Hence, proved that sin 60° cos 30° - cos 60° sin 30° = sin 30°.

(ii) Left Hand Side :

2 sin 30° cos 30°

= 2×12×322\times\dfrac{1}{2} \times \dfrac{\sqrt{3}}{2}

= 32\dfrac{\sqrt{3}}{2}

Right Hand Side :

sin 60° = 32\dfrac{\sqrt{3}}{2}

Hence, proved that 2 sin 30° cos 30° = sin 60°.

(iii) Left Hand Side :

2 sin 45° cos 45°

= 2×12×122\times\dfrac{1}{\sqrt2} \times \dfrac{1}{\sqrt2}

= 1

Right Hand Side :

sin 90° = 1

Hence, proved that 2 sin 45° cos 45° = sin 90°.

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