Simplify sin(9x)/cot(12x).

Step-by-step solution

1. 1

   Replace the cotangent by its definition. The quotient identity is

   $\cot\theta=\frac{\cos\theta}{\sin\theta},$

   so with $\theta=12x$ the expression becomes

   $\frac{\sin(9x)}{\dfrac{\cos(12x)}{\sin(12x)}}.$

   Rewriting in terms of sine and cosine is almost always the right first move when a reciprocal trig function sits in an awkward place.

2. 2

   Divide by a fraction by multiplying by its reciprocal. A compound fraction $\dfrac{A}{B/C}$ equals $A\cdot\dfrac{C}{B}$:

   $\frac{\sin(9x)}{\dfrac{\cos(12x)}{\sin(12x)}}=\sin(9x)\cdot\frac{\sin(12x)}{\cos(12x)}.$

   The classic error is to write $\sin(9x)\cdot\frac{\cos(12x)}{\sin(12x)}$ — dividing by $\cot$ is the same as multiplying by $\tan$, not by $\cot$.

3. 3

   Recognise the tangent. Since $\dfrac{\sin\theta}{\cos\theta}=\tan\theta$,

   $\frac{\sin(9x)}{\cot(12x)}=\sin(9x)\tan(12x).$

   Equivalently the answer can be left as $\dfrac{\sin(9x)\sin(12x)}{\cos(12x)}$; both forms are correct, and the $\tan$ version is the more compact one.

4. 4

   State the domain restrictions. The original expression requires $\cot(12x)$ to be defined and nonzero, so $\sin(12x)\ne0$ and $\cos(12x)\ne0$, i.e. $12x\ne\frac{k\pi}{2}$. The simplified form $\sin(9x)\tan(12x)$ only needs $\cos(12x)\ne0$, so it is defined at slightly more points than the original — the two agree wherever both make sense.

5. 5

   Check numerically. At $x=0.1$: $\sin(0.9)=0.783327$ and $\cot(1.2)=0.3887796$, so the original quotient is $\frac{0.783327}{0.3887796}=2.014836$; and $\sin(0.9)\tan(1.2)=0.783327\times2.572152=2.014836$ ✓. Agreement at a value where neither function is near a pole confirms the algebra.