Answer :

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.