What is the value of √3 cosec 20° – sec 20° ?

1.	What is the value of √3 cosec 20° – sec 20° ?
Answer» √3 cosec 20° – sec 20° = \(\frac{\sqrt3}{sin\,20°}\) - \(\frac{1}{cos\,20°}\) = \(\frac{\sqrt3\,cos\,20°-sin\,20°}{sin20°\,cos20°}\) = \(\frac{4}{2\,sin20°\,cos20°}\bigg(\frac{\sqrt3}{2}cos\,20°-\frac12sin\,20°\bigg)\) (Multiplying numerator and denominator by 4) = \(\frac{4}{sin\,40°}\) (sin 60° cos 20° – cos 60° sin 20°) (∵ sin 2θ = 2 sin θ cos θ) = \(\frac{4}{sin\,40°}\) (sin (60° – 20°)) (∵ sin (A – B) = sin A cos B – cos A sin B) = \(\frac{4}{sin\,40°}\). sin 40° = 4.

Answer»

√3 cosec 20° – sec 20°

= \(\frac{\sqrt3}{sin\,20°}\) - \(\frac{1}{cos\,20°}\) = \(\frac{\sqrt3\,cos\,20°-sin\,20°}{sin20°\,cos20°}\)

= \(\frac{4}{2\,sin20°\,cos20°}\bigg(\frac{\sqrt3}{2}cos\,20°-\frac12sin\,20°\bigg)\) (Multiplying numerator and denominator by 4)

= \(\frac{4}{sin\,40°}\) (sin 60° cos 20° – cos 60° sin 20°) (∵ sin 2θ = 2 sin θ cos θ)

= \(\frac{4}{sin\,40°}\) (sin (60° – 20°)) (∵ sin (A – B) = sin A cos B – cos A sin B)

= \(\frac{4}{sin\,40°}\). sin 40° = 4.

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