\( \operatorname { Cos } ( 90 ^ { \circ } + x ) \cdot \tan ( 540 ^ { \circ } + x ) \cdot \operatorname { Cos } ( 180 ^ { \circ } - x ) + \operatorname { Sin } ( - 90 ^ { \circ } ) \)

Let's simplify the expression step by step: The expression is: \[ \operatorname{Cos}(90^{\circ} + x) \cdot \tan(540^{\circ} + x) \cdot \operatorname{Cos}(180^{\circ} - x) + \operatorname{Sin}(-90^{\circ}) \] ### Step 1: Simplify each trigonometric function 1. **Cosine of \(90^{\circ} + x\)**: \[ \operatorname{Cos}(90^{\circ} + x) = -\operatorname{Sin}(x) \] 2. **Tangent of \(540^{\circ} + x\)**: \[ 540^{\circ} = 360^{\circ} + 180^{\circ} \implies \tan(540^{\circ} + x) = \tan(180^{\circ} + x) = \tan(x) \] 3. **Cosine of \(180^{\circ} - x\)**: \[ \operatorname{Cos}(180^{\circ} - x) = -\operatorname{Cos}(x) \] 4. **Sine of \(-90^{\circ}\)**: \[ \operatorname{Sin}(-90^{\circ}) = -1 \] ### Step 2: Substitute back into the expression Now substituting these results back into the original expression: \[ (-\operatorname{Sin}(x)) \cdot \tan(x) \cdot (-\operatorname{Cos}(x)) + (-1) \] This simplifies to: \[ \operatorname{Sin}(x) \cdot \tan(x) \cdot \operatorname{Cos}(x) - 1 \] ### Step 3: Further simplify Recall that: \[ \tan(x) = \frac{\operatorname{Sin}(x)}{\operatorname{Cos}(x)} \] Thus: \[ \operatorname{Sin}(x) \cdot \tan(x) \cdot \operatorname{Cos}(x) = \operatorname{Sin}(x) \cdot \frac{\operatorname{Sin}(x)}{\operatorname{Cos}(x)} \cdot \operatorname{Cos}(x) = \operatorname{Sin}^2(x) \] ### Final Expression Putting it all together, we have: \[ \operatorname{Sin}^2(x) - 1 \] This can be expressed as: \[ -\operatorname{Cos}^2(x) \] Thus, the final simplified expression is: \[ -\operatorname{Cos}^2(x) \]