$$ \sin 210 ^ { \circ } \cdot \sin \frac { 225 ^ { \circ } \cdot \operatorname { cog } ( heta - 180 ^ { \circ } ) } { \sin 135 ^ { \circ } \cdot \operatorname { cog } heta } $$

Question

$$ \sin 210 ^ { \circ } \cdot \sin \frac { 225 ^ { \circ } \cdot \operatorname { cog } ( 	heta - 180 ^ { \circ } ) } { \sin 135 ^ { \circ } \cdot \operatorname { cog } 	heta } $$

UpStudy AI · Accepted Answer

We start with the expression
$$
\sin 210^\circ \cdot \sin \frac{225^\circ \cdot \operatorname{cot}(\theta-180^\circ)}{\sin 135^\circ \cdot \operatorname{cot}\theta}.
$$

**Step 1. Evaluate the sine values.**

Recall:
$$
\sin 210^\circ = \sin(180^\circ+30^\circ) = -\sin 30^\circ = -\frac{1}{2},
$$
$$
\sin 225^\circ = \sin(180^\circ+45^\circ) = -\sin 45^\circ = -\frac{\sqrt{2}}{2},
$$
$$
\sin 135^\circ = \sin(180^\circ-45^\circ) = \sin 45^\circ = \frac{\sqrt{2}}{2}.
$$

**Step 2. Substitute the numerical values.**

The expression becomes
$$
-\frac{1}{2} \cdot \sin \frac{\left(-\frac{\sqrt{2}}{2}\right) \cdot \operatorname{cot}(\theta-180^\circ)}{\left(\frac{\sqrt{2}}{2}\right) \cdot \operatorname{cot}\theta}.
$$

**Step 3. Simplify the fraction inside the sine.**

Notice that the factors $$\frac{\sqrt{2}}{2}$$ cancel:
$$
\frac{\left(-\frac{\sqrt{2}}{2}\right) \cdot \operatorname{cot}(\theta-180^\circ)}{\left(\frac{\sqrt{2}}{2}\right) \cdot \operatorname{cot}\theta}
=\frac{-\operatorname{cot}(\theta-180^\circ)}{\operatorname{cot}\theta}.
$$
The overall expression is now
$$
-\frac{1}{2} \cdot \sin \left(-\frac{\operatorname{cot}(\theta-180^\circ)}{\operatorname{cot}\theta}\right).
$$

**Step 4. Use the odd property of sine.**

Since $$\sin(-x) = -\sin(x)$$,
$$
-\frac{1}{2} \cdot \sin \left(-\frac{\operatorname{cot}(\theta-180^\circ)}{\operatorname{cot}\theta}\right)
= -\frac{1}{2} \cdot \left(-\sin \frac{\operatorname{cot}(\theta-180^\circ)}{\operatorname{cot}\theta}\right)
= \frac{1}{2} \sin \frac{\operatorname{cot}(\theta-180^\circ)}{\operatorname{cot}\theta}.
$$

**Step 5. Use the periodicity of the cotangent function.**

The cotangent has a period of $$180^\circ$$, so
$$
\operatorname{cot}(\theta-180^\circ)=\operatorname{cot}\theta.
$$
Thus,
$$
\frac{\operatorname{cot}(\theta-180^\circ)}{\operatorname{cot}\theta} = \frac{\operatorname{cot}\theta}{\operatorname{cot}\theta} = 1.
$$

**Step 6. Write the final simplified expression.**

Substitute this result into the expression:
$$
\frac{1}{2} \sin 1.
$$

The final answer is
$$
\frac{1}{2}\sin 1.
$$
The simplified expression is $$ \frac{1}{2} \sin 1 $$.

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