To solve $\sin(105^\circ) + \cos(105^\circ)$, we can break it down using trigonometric identities.

Step 1: Use sum identities

The sum of sine and cosine functions can be simplified using the following identity:

$\sin x + \cos x = \sqrt{2} \sin\left(x + 45^\circ\right)$

So, let's apply this to $x = 105^\circ$:

$\sin(105^\circ) + \cos(105^\circ) = \sqrt{2} \sin(105^\circ + 45^\circ)$

This simplifies to:

$\sin(105^\circ) + \cos(105^\circ) = \sqrt{2} \sin(150^\circ)$

Step 2: Evaluate $\sin(150^\circ)$

Now, recall that $\sin(150^\circ) = \sin(30^\circ) = \frac{1}{2}$.

Thus:

$\sin(105^\circ) + \cos(105^\circ) = \sqrt{2} \times \frac{1}{2} = \frac{\sqrt{2}}{2}$

Final Answer:

$\sin(105^\circ) + \cos(105^\circ) = \frac{\sqrt{2}}{2}$

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Would you like more details on this solution? Here are 5 related questions you might find helpful:

1. What is the exact value of $\sin(105^\circ)$ and $\cos(105^\circ)$?
2. How can I apply the sum-to-product identities for trigonometric functions?
3. How do the values of sine and cosine behave for angles greater than $90^\circ$?
4. Can this method be used for angles other than 105°? How?
5. How can I use a calculator to verify this result?

Tip: Using sum or difference identities for trigonometric functions simplifies calculations, especially when dealing with non-standard angles like 105°.