The value of csc 10^circ - sqrt3 sec 10^circ$\csc 10^{\circ} - \sqrt{3} \sec 10^{\circ}$ is equal to:

### Related Formula sin(A - B) = sin A cos B - cos A sin B$$\sin(A - B) = \sin A \cos B - \cos A \sin B$$ sin 2A = 2 sin A cos A$$\sin 2A = 2 \sin A \cos A$$ ### Core Logic csc 10^circ - sqrt3 sec 10^circ = frac1sin 10^circ - fracsqrt3cos 10^circ$$\csc 10^{\circ} - \sqrt{3} \sec 10^{\circ} = \frac{1}{\sin 10^{\circ}} - \frac{\sqrt{3}}{\cos 10^{\circ}}$$ = fraccos 10^circ - sqrt3sin 10^circsin 10^circ cos 10^circ$$= \frac{\cos 10^{\circ} - \sqrt{3}\sin 10^{\circ}}{\sin 10^{\circ} \cos 10^{\circ}}$$ ### Step 1: Sine Transformation Multiply and divide the numerator by 2 to inject standard trig values: = frac2 left(frac12cos 10^circ - fracsqrt32sin 10^circright)sin 10^circ cos 10^circ$$= \frac{2 \left(\frac{1}{2}\cos 10^{\circ} - \frac{\sqrt{3}}{2}\sin 10^{\circ}\right)}{\sin 10^{\circ} \cos 10^{\circ}}$$ Substitute sin 30^circ = 1/2$\sin 30^{\circ} = 1/2$ and cos 30^circ = sqrt3/2$\cos 30^{\circ} = \sqrt{3}/2$: = frac2 (sin 30^circ cos 10^circ - cos 30^circ sin 10^circ)sin 10^circ cos 10^circ$$= \frac{2 (\sin 30^{\circ} \cos 10^{\circ} - \cos 30^{\circ} \sin 10^{\circ})}{\sin 10^{\circ} \cos 10^{\circ}}$$ ### Step 2: Apply Multiple Angle identities The numerator becomes 2 sin(30^circ - 10^circ) = 2 sin 20^circ$2 \sin(30^{\circ} - 10^{\circ}) = 2 \sin 20^{\circ}$. Multiply and divide the denominator by 2 to construct sin 2theta$\sin 2\theta$: = frac4 sin 20^circ2 sin 10^circ cos 10^circ$$= \frac{4 \sin 20^{\circ}}{2 \sin 10^{\circ} \cos 10^{\circ}}$$ = frac4 sin 20^circsin 20^circ$$= \frac{4 \sin 20^{\circ}}{\sin 20^{\circ}}$$ = 4$= 4$ ### Pattern Recognition Any expression structured as A csc theta - B sec theta$A \csc \theta - B \sec \theta$ instantly signals a fraction collapse to 2 sin(alpha - theta) / sin(2theta)$2 \sin(\alpha - \theta) / \sin(2\theta)$ via multiplication by 2. When constants are 1$1$ and sqrt3$\sqrt{3}$, the anchor is always 30^circ$30^{\circ}$ or 60^circ$60^{\circ}$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Maths: Trigonometric Functions