`lim_(ntooo)(sin^(n)1+cos^(n)1)^(n)` is equal to

Answer

To solve the limit \( \lim_{n \to \infty} \left( \sin^n(1) + \cos^n(1) \right)^n \), we will follow these steps:

### Step 1: Identify the values of \(\sin(1)\) and \(\cos(1)\)

First, we need to evaluate \(\sin(1)\) and \(\cos(1)\). Since \(1\) radian is approximately \(57.3^\circ\), we know that both \(\sin(1)\) and \(\cos(1)\) are positive values less than \(1\).

### Step 2: Analyze \(\sin^n(1)\) and \(\cos^n(1)\)

As \(n\) approaches infinity, both \(\sin^n(1)\) and \(\cos^n(1)\) will approach \(0\) because any number between \(0\) and \(1\) raised to the power of \(n\) (as \(n\) approaches infinity) tends to \(0\).

### Step 3: Combine the terms

Now we can express the limit:
\[ \sin^n(1) + \cos^n(1) \to 0 + 0 = 0 \]

### Step 4: Raise the sum to the power of \(n\)

Next, we consider the expression:
\[ \left( \sin^n(1) + \cos^n(1) \right)^n \]
As \(n\) approaches infinity, since \(\sin^n(1) + \cos^n(1) \to 0\), we have:
\[ \left( \sin^n(1) + \cos^n(1) \right)^n \to 0^n \]
which is \(0\) for any positive \(n\).

### Step 5: Conclude the limit

Thus, we conclude that:
\[ \lim_{n \to \infty} \left( \sin^n(1) + \cos^n(1) \right)^n = 0 \]

### Final Answer
The limit is:
\[ \boxed{0} \]