To solve the trigonometric equation [tex]\(-\frac{\operatorname{sen} x}{\csc x} = \cos^2 x\)[/tex], let's go through the steps methodically.
1. Understand the trigonometric terms:
[tex]\(\operatorname{sen} x\)[/tex] is the same as [tex]\(\sin x\)[/tex].
[tex]\(\csc x\)[/tex] is the cosecant of [tex]\(x\)[/tex], which is [tex]\(\frac{1}{\sin x}\)[/tex].
2. Rewrite the given equation with these terms:
[tex]\[
-\frac{\sin x}{\csc x} = \cos^2 x
\][/tex]
3. Recall that [tex]\(\csc x = \frac{1}{\sin x}\)[/tex]:
Substitute this into the equation:
[tex]\[
-\frac{\sin x}{\frac{1}{\sin x}} = \cos^2 x
\][/tex]
4. Simplify the left-hand side of the equation:
[tex]\[
-\sin x \times \sin x = \cos^2 x
\][/tex]
Which simplifies to:
[tex]\[
-\sin^2 x = \cos^2 x
\][/tex]
5. Use the Pythagorean identity:
The Pythagorean identity states:
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
So, we can express [tex]\(\sin^2 x\)[/tex] in terms of [tex]\(\cos^2 x\)[/tex]:
[tex]\[
\sin^2 x = 1 - \cos^2 x
\][/tex]
6. Substitute [tex]\(\sin^2 x\)[/tex] into the simplified equation:
[tex]\[
- (1 - \cos^2 x) = \cos^2 x
\][/tex]
Simplify this further:
[tex]\[
-1 + \cos^2 x = \cos^2 x
\][/tex]
Combine like terms:
[tex]\[
-1 = 2 \cos^2 x
\][/tex]
7. Solve for [tex]\(\cos^2 x\)[/tex]:
[tex]\[
\cos^2 x = -\frac{1}{2}
\][/tex]
8. Analyze the result:
The expression [tex]\(\cos^2 x = -\frac{1}{2}\)[/tex] is problematic because [tex]\(\cos^2 x\)[/tex] represents the square of the cosine function, which is always a non-negative value (since squaring any real number, whether positive or negative or zero, results in a non-negative number). The range of [tex]\(\cos^2 x\)[/tex] is [tex]\([0, 1]\)[/tex], meaning it cannot be negative.
Thus:
[tex]\[
\cos^2 x = -\frac{1}{2}
\][/tex]
Isn't possible for any real [tex]\(x\)[/tex].
Therefore, the given trigonometric equation [tex]\(-\frac{\operatorname{sen} x}{\csc x} = \cos^2 x\)[/tex] has no real solution.
