Let's analyze and solve the given expression step-by-step:
We need to evaluate the expression:
[tex]\[
\cot (0^\circ + 90^\circ) = \frac{\cot 0^\circ \times \cot 90^\circ - 1}{\cot 90^\circ + \cot 0^\circ}
\][/tex]
### Step 1: Evaluate [tex]\(\cot 0^\circ\)[/tex]
[tex]\(\cot\)[/tex] is defined as the reciprocal of [tex]\(\tan\)[/tex]. Therefore:
[tex]\[
\cot 0^\circ = \frac{1}{\tan 0^\circ}
\][/tex]
However, [tex]\(\tan 0^\circ = 0\)[/tex], so:
[tex]\[
\cot 0^\circ = \frac{1}{0}
\][/tex]
But division by zero is undefined, so [tex]\(\cot 0^\circ\)[/tex] is undefined.
### Step 2: Evaluate [tex]\(\cot 90^\circ\)[/tex]
Similarly, we look at:
[tex]\[
\cot 90^\circ = \frac{1}{\tan 90^\circ}
\][/tex]
But [tex]\(\tan 90^\circ\)[/tex] itself is undefined (it tends to infinity), which means:
[tex]\[
\cot 90^\circ = 0
\][/tex]
### Step 3: Substitute the values into the expression
Substitute the values we have:
[tex]\(\cot 0^\circ\)[/tex] is undefined, often represented as approaching infinity ([tex]\(\infty\)[/tex]).
[tex]\(\cot 90^\circ = 0\)[/tex]
Now, substitute these into the given expression:
[tex]\[
\cot (0^\circ + 90^\circ) = \frac{\cot 0^\circ \times \cot 90^\circ - 1}{\cot 90^\circ + \cot 0^\circ} = \frac{\infty \times 0 - 1}{0 + \infty}
\][/tex]
### Step 4: Simplify the expression
Attempt to simplify the numerator and the denominator:
1. Numerator: [tex]\(\infty \times 0\)[/tex] is indeterminate, but let's consider each part:
- Since [tex]\(\cot 0^\circ\)[/tex] is undefined and considered as approaching infinity, [tex]\(\infty \times 0\)[/tex] is undefined due to the indeterminate form.
- Therefore the numerator [tex]\( \infty \times 0 - 1 \)[/tex] cannot be simplified straightforwardly and is undefined.
2. Denominator: [tex]\(0 + \infty\)[/tex] is [tex]\(\infty\)[/tex], since addition of any finite number to infinity results in infinity.
Thus, we have:
[tex]\[
\cot (90^\circ) = \frac{\text{undefined (}\infty \times 0 - 1\text{)}}{\infty}
\][/tex]
### Step 5: Conclusion
The indeterminate form in the numerator combined with an infinite denominator results in the expression being undefined.
Therefore, we can conclude:
[tex]\[
\cot (90^\circ) = \frac{\text{undefined}}{\infty} \implies \text{undefined} \, (\text{or nan})
\][/tex]
The values for [tex]\(\cot 0^\circ\)[/tex] and [tex]\(\cot 90^\circ\)[/tex] are [tex]\( \infty \)[/tex] and [tex]\( 0 \)[/tex] respectively, and the final evaluation of [tex]\(\cot(0^\circ + 90^\circ)\)[/tex] is also undefined.
