Answer
The reduction formula is $$\cot\theta$$
Work Step by Step
*Summary of the method:
For a formula $f(Q\pm\theta)$
1) See that $Q$ terminates on the $x$ or $y$ axis. If it terminates on the $x$ axis, go for Case 1. If it terminates on the $y$ axis, go for Case 2.
2) Case 1:
- For a small positive value of $\theta$, determinate $Q\pm\theta$ lies in which quadrant.
- If $f\gt0$, use a $+$ sign. If $f\lt0$, use a $-$ sign.
- The reduced form will have that sign, $f$ the function and $\theta$ the angle.
3) Case 2:
- For a small positive value of $\theta$, determinate $Q\pm\theta$ lies in which quadrant.
- If $f\gt0$, use a $+$ sign. If $f\lt0$, use a $-$ sign.
- The reduced form will have that sign, cofunction of$f$ as the function and $\theta$ the angle.
$$\tan(270^\circ-\theta)$$
1) $270^\circ$ terminates on the $y$ axis. We go for Case 2.
2) As $\theta$ is a very small positive value, which means $\theta\gt0$, $$270^\circ\gt(270^\circ-\theta)\gt180^\circ$$
So $270^\circ-\theta$ lies in quadrant III.
3) Tangent is positive in quadrant III. So we use a $+$ sign.
4) In case 2, we use the cofunction of the given formula as the function. That means the given formula is tangent, so the reduced form would have cotangent as the function, combined with the positive sign proved above. Overall, the reduced form would be
$$\cot\theta$$
