Without using a calculator, fully simplify the expression below:

(2sin165°.cos345°)/(cos45°cos15°+sin45°sin15°)

Let's simplify the given expression step by step:

\[
\text{Expression:} \quad \frac{2 \sin 165^\circ \cdot \cos 345^\circ}{\cos 45^\circ \cos 15^\circ + \sin 45^\circ \sin 15^\circ}
\]

### Step 1: Simplify the denominator
Notice that the denominator is in the form of a trigonometric identity:

\[
\cos 45^\circ \cos 15^\circ + \sin 45^\circ \sin 15^\circ
\]

This expression matches the cosine addition formula:

\[
\cos(x - y) = \cos x \cos y + \sin x \sin y
\]

Here, \(x = 45^\circ\) and \(y = 15^\circ\):

\[
\cos 45^\circ \cos 15^\circ + \sin 45^\circ \sin 15^\circ = \cos(45^\circ - 15^\circ) = \cos 30^\circ
\]

So, the denominator simplifies to \(\cos 30^\circ\).

### Step 2: Simplify the numerator
Now, simplify the numerator \(2 \sin 165^\circ \cdot \cos 345^\circ\).

Note the angle relationships:
- \(\sin 165^\circ = \sin(180^\circ - 15^\circ) = \sin 15^\circ\)
- \(\cos 345^\circ = \cos(360^\circ - 15^\circ) = \cos 15^\circ\)

So the numerator becomes:

\[
2 \sin 165^\circ \cdot \cos 345^\circ = 2 \sin 15^\circ \cdot \cos 15^\circ
\]

Using the double angle identity for sine:

\[
\sin(2x) = 2 \sin x \cos x
\]

Here, \(x = 15^\circ\):

\[
2 \sin 15^\circ \cdot \cos 15^\circ = \sin 30^\circ
\]

So, the numerator simplifies to \(\sin 30^\circ\).

### Step 3: Combine the results
Now, the entire expression simplifies to:

\[
\frac{\sin 30^\circ}{\cos 30^\circ}
\]

This is the definition of \(\tan 30^\circ\):

\[
\tan 30^\circ = \frac{1}{\sqrt{3}}
\]

### Final Answer:
The simplified value of the expression is:

\[
\frac{1}{\sqrt{3}}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the expression change if the angles were different, say \( \sin 150^\circ \) and \( \cos 330^\circ \)?
2. Can you use the same method to simplify an expression with different trigonometric identities, like \( \sin(x+y) \)?
3. What is the relationship between the angles \(165^\circ\) and \(345^\circ\) in terms of trigonometric identities?
4. How would you simplify the expression if you were given \( \sin 135^\circ \) and \( \cos 45^\circ \) instead?
5. How can the double-angle identity for cosine be used in similar expressions?

**Tip:** Understanding trigonometric identities like sum-to-product and angle subtraction is crucial for simplifying complex trigonometric expressions.

Learn how to simplify the trigonometric expression (2sin165°.cos345°)/(cos45°cos15°+sin45°sin15°) step-by-step without using a calculator. This problem covers trigonometric identities, including sum-to-product and double angle identities, suitable for Grades 10-12 students.

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