Math Snap

STEP 1

1. The equation involves trigonometric functions: tangent, cosine, and sine.
2. We will use trigonometric identities to simplify and solve the equation.
3. The solution will be found within the interval $0^{\circ} \leq x \leq 360^{\circ}$.

STEP 2

1. Use trigonometric identities to simplify the equation.
2. Solve the simplified equation for $x$.
3. Verify solutions within the given interval.

STEP 3

Start by using the identity $\tan x = \frac{\sin x}{\cos x}$ to rewrite the equation:
$$\tan x \cos x \sin x - 1 = 0$$ Substitute $\tan x = \frac{\sin x}{\cos x}$:
$$\frac{\sin x}{\cos x} \cos x \sin x - 1 = 0$$ Simplify the expression:
$$\sin^2 x - 1 = 0$$

STEP 4

Solve the equation $\sin^2 x - 1 = 0$:
$$\sin^2 x = 1$$ Take the square root of both sides:
$$\sin x = \pm 1$$

Determine the angles $x$ where $\sin x = 1$ and $\sin x = -1$ within the interval $0^{\circ} \leq x \leq 360^{\circ}$:
- $\sin x = 1$ at $x = 90^{\circ}$
- $\sin x = -1$ at $x = 270^{\circ}$
Thus, the solutions are:
$$x = 90^{\circ}, 270^{\circ}$$ The values of $x$ are:
$$\boxed{90^{\circ}, 270^{\circ}}$$