JEE Main 2025 Jan 23 Shift 1, Mathematics Q15: Trigonometric Ratios And Identities

Combine the bracket over a single denominator:

$\cot 10^{\circ }\cot 70^{\circ }-1=\frac{\cos 10^{\circ }\cos 70^{\circ }-\sin 10^{\circ }\sin 70^{\circ }}{\sin 10^{\circ }\sin 70^{\circ }}=\frac{\cos (10^{\circ }+70^{\circ })}{\sin 10^{\circ }\sin 70^{\circ }}=\frac{\cos 80^{\circ }}{\sin 10^{\circ }\sin 70^{\circ }}$.

Since $\cos 80^{\circ }=\sin 10^{\circ }$, this simplifies to $\frac{1}{\sin 70^{\circ }}$.

Multiplying by $\sin 70^{\circ }$ gives $1$.