Simplify the expression:
[tex]\[ (\cos \theta) \cot \theta + \sin \theta = \sec \theta \][/tex]



Format the following question or task so that it is easier to read.
Fix any grammar or spelling errors.
Remove phrases that are not part of the question.
Do not remove or change LaTeX formatting.
Do not change or remove [tex] [/tex] tags.
If the question is nonsense, rewrite it so that it makes sense.
-----
The rate of formation of NOBr from the decomposition of NOBr is found to be rate = k[NOBr]. This reaction is
A) First order B) Second order C) Third order D) Zero order
-----

Response:
The rate of formation of NOBr from the decomposition of NOBr is given by:

[tex]\[
\text{rate} = k[\text{NOBr}]
\][/tex]

This reaction is:
A. First order
B. Second order
C. Third order
D. Zero order



Answer :

Sure, let's work through solving the trigonometric equation [tex]\((\cos \theta) \cot \theta + \sin \theta = \sec \theta\)[/tex].

First, recall the definitions in terms of sine and cosine:
- [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex]
- [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex]

Substitute these identities into the equation:

[tex]\[ (\cos \theta) \left(\frac{\cos \theta}{\sin \theta}\right) + \sin \theta = \frac{1}{\cos \theta} \][/tex]

This simplifies to:

[tex]\[ \frac{\cos^2 \theta}{\sin \theta} + \sin \theta = \frac{1}{\cos \theta} \][/tex]

We need a common denominator to combine the terms on the left-hand side. The common denominator will be [tex]\(\sin \theta\)[/tex]:

[tex]\[ \frac{\cos^2 \theta + \sin^2 \theta \cdot \sin \theta}{\sin \theta} = \frac{1}{\cos \theta} \][/tex]

Recognize that [tex]\(\cos^2 \theta + \sin^2 \theta = 1\)[/tex] by the Pythagorean identity, so:

[tex]\[ \frac{1}{\sin \theta} = \frac{1}{\cos \theta} \][/tex]

Set the equivalent numerators equal:

[tex]\[ \sin \theta = \cos \theta \][/tex]

We solve for [tex]\(\theta\)[/tex] where [tex]\(\sin \theta = \cos \theta\)[/tex]. This occurs when [tex]\(\theta\)[/tex] results in an angle where their sines and cosines are equal. One known solution occurs at:

[tex]\[ \theta = \frac{\pi}{4} \][/tex]

However, we should consider the periodic nature of trigonometric functions. For cosine and sine to be equal, the angles can also differ by [tex]\(\pi\)[/tex]. Thus, other solutions are in the form:

[tex]\[ \theta = \frac{\pi}{4} + k\pi \quad \text{where } k \text{ is any integer} \][/tex]

We are particularly interested in specific solutions within the main cycle of [tex]\([0, 2\pi)\)[/tex]. By examining [tex]\(k\pi\)[/tex] shifts:

[tex]\[ \theta = \frac{\pi}{4} \quad \text{and} \quad \theta = \frac{\pi}{4} + \pi = \frac{\pi}{4} + \frac{4\pi}{4} = \frac{5\pi}{4} \][/tex]

Note that angles can be expressed negatively as well through cycle shifting, producing:

[tex]\[ \theta = -\frac{3\pi}{4} \quad (\text{another form}) \][/tex]

Therefore, the solutions within one cycle [tex]\([0, 2\pi)\)[/tex] for the given equation are:

[tex]\[ \theta = \frac{\pi}{4} \quad \text{and} \quad \theta = \frac{5\pi}{4} \][/tex]

But including the negative form, we get:

[tex]\[ \theta = \frac{\pi}{4} \quad \text{and} \quad \theta = -\frac{3\pi}{4} \][/tex]

So the final solutions are:

[tex]\[ \theta = \frac{\pi}{4} \quad \text{and} \quad \theta = -\frac{3\pi}{4} \][/tex]