The value of cosec10°– √3sec10°

The correct answer is 4.

To find the value of the trigonometric expression:
$\mathrm{cosec}10°-\sqrt{3}\sec 10°$
we can convert the cosecant and secant functions into sine and cosine functions.

Using the reciprocal identities, $\mathrm{cosec}\theta =\frac{1}{\sin \theta }$ and $\sec \theta =\frac{1}{\cos \theta }$:
$\mathrm{cosec}10°-\sqrt{3}\sec 10°=\frac{1}{\sin 10°}-\frac{\sqrt{3}}{\cos 10°}$

Taking the common denominator, we get:
$=\frac{\cos 10°���\sqrt{3}\sin 10°}{\sin 10°\cos 10°}$

Multiply and divide the numerator by 2:
$=\frac{2\left(\frac{1}{2}\cos 10°-\frac{\sqrt{3}}{2}\sin 10°\right)}{\sin 10°\cos 10°}$

We know that $\sin 30°=\frac{1}{2}$ and $\cos 30°=\frac{\sqrt{3}}{2}$. Substituting these values in the numerator:
$=\frac{2\left(\sin 30°\cos 10°-\cos 30°\sin 10°\right)}{\sin 10°\cos 10°}$

Apply the trigonometric difference identity $\sin (A-B)=\sin A\cos B-\cos A\sin B$:
$=\frac{2\sin \left(30°-10°\right)}{\sin 10°\cos 10°}$
$=\frac{2\sin 20°}{\sin 10°\cos 10°}$

Multiply both the numerator and the denominator by 2 to use the double-angle identity $2\sin \theta \cos \theta =\sin 2\theta$:
$=\frac{2\times 2\sin 20°}{2\sin 10°\cos 10°}$
$=\frac{4\sin 20°}{\sin 20°}$
$=4$