Certainly! Let's walk through each of the given expressions step-by-step to understand and confirm the results.
### 1.1.1 [tex]\(\cos x + \cos y\)[/tex]
This is a basic trigonometric expression involving the sum of cosines of two different variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
Given:
[tex]\[ \cos x + \cos y \][/tex]
There is no further simplification needed for this expression. The result is simply:
[tex]\[ \cos x + \cos y \][/tex]
### 1.1.2 [tex]\(2 \tan^2(2x + 3y)\)[/tex]
This expression involves the tangent function squared, scaled by a factor of 2.
Given:
[tex]\[ 2 \tan^2(2x + 3y) \][/tex]
This is the tangent function of the angle [tex]\(2x + 3y\)[/tex] squared, multiplied by 2. Again, no further simplification is required. The result is:
[tex]\[ 2 \tan^2(2x + 3y) \][/tex]
### 1.1.3 [tex]\(4 \sin \left( \frac{y - x}{4} \right)\)[/tex]
This expression involves the sine function scaled by a factor of 4. Notice that it is a sine function of a fractional angle.
Given:
[tex]\[ 4 \sin \left( \frac{y - x}{4} \right) \][/tex]
We can reformat the argument inside the sine function for clarity:
[tex]\[ 4 \sin \left( \frac{1}{4} (y - x) \right) \][/tex]
Alternatively, this can also be written as:
[tex]\[ -4 \sin \left( \frac{x - y}{4} \right) \][/tex]
This acknowledges the odd nature of the sine function, indicating that:
[tex]\[ \sin(-\theta) = -\sin(\theta) \][/tex]
Therefore:
[tex]\[ 4 \sin \left( \frac{1}{4} (y - x) \right) = -4 \sin \left( \frac{x - y}{4} \right) \][/tex]
Thus, the simplified result is:
[tex]\[ -4 \sin \left( \frac{x}{4} - \frac{y}{4} \right) \][/tex]
So, our final results are:
1. [tex]\(\cos x + \cos y\)[/tex]
2. [tex]\(2 \tan^2(2x + 3y)\)[/tex]
3. [tex]\(-4 \sin \left( \frac{x}{4} - \frac{y}{4} \right)\)[/tex]
Each of these correctly represents the given trigonometric expressions.
