$$ \left. \begin{array} { l } { \frac { 3 ) \cos ^ { 4 } 90 ^ { \circ } - \sin ^ { 4 } ( - 10 ^ { \circ } ) } { \cos 55 ^ { \circ } \cos 325 ^ { \circ } } } \ { \frac { \cos ( 40 ^ { \circ } - x ) \cos x - \sin ( 40 ^ { \circ } - x ) \sin x } { \sin 2 \cos ^ { \circ } \cos 155 ^ { \circ } } = \pi } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { \frac { 3 ) \cos ^ { 4 } 90 ^ { \circ } - \sin ^ { 4 } ( - 10 ^ { \circ } ) } { \cos 55 ^ { \circ } \cos 325 ^ { \circ } } } \ { \frac { \cos ( 40 ^ { \circ } - x ) \cos x - \sin ( 40 ^ { \circ } - x ) \sin x } { \sin 2 \cos ^ { \circ } \cos 155 ^ { \circ } } = \pi } \end{array} ight. $$

UpStudy AI · Accepted Answer

Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\frac{\left(\cos\left(40-x\right)\cos\left(x\right)-\sin\left(40-x\right)\sin\left(x\right)\right)}{\left(\sin\left(2\right)\cos\left(155\right)\right)}=\pi $$
- step1: Remove the parentheses:
  $$\frac{\cos\left(40-x\right)\cos\left(x\right)-\sin\left(40-x\right)\sin\left(x\right)}{\sin\left(2\right)\cos\left(155\right)}=\pi $$
- step2: Simplify:
  $$\frac{\cos\left(40\right)}{\sin\left(2\right)\cos\left(155\right)}=\pi $$
- step3: Multiply both sides of the equation by LCD:
  $$\frac{\cos\left(40\right)}{\sin\left(2\right)\cos\left(155\right)}\times \sin\left(2\right)\cos\left(155\right)=\pi \sin\left(2\right)\cos\left(155\right)$$
- step4: Simplify the equation:
  $$\cos\left(40\right)=\pi \sin\left(2\right)\cos\left(155\right)$$
- step5: Calculate:
  $$-0.666938=\pi \sin\left(2\right)\cos\left(155\right)$$
- step6: Calculate:
  $$-0.666938=-1.391646$$
- step7: The statement is false:
  $$x \in \varnothing $$
Calculate or simplify the expression $$ (3 * \cos(90)^4 - \sin(-10)^4) / (\cos(55) * \cos(325)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(3\cos\left(90^{4}\right)-\sin\left(\left(-10\right)^{4}\right)\right)}{\left(\cos\left(55\right)\cos\left(325\right)\right)}$$
- step1: Remove the parentheses:
  $$\frac{3\cos\left(90^{4}\right)-\sin\left(\left(-10\right)^{4}\right)}{\cos\left(55\right)\cos\left(325\right)}$$
- step2: Evaluate the power:
  $$\frac{3\cos\left(90^{4}\right)-\sin\left(10000\right)}{\cos\left(55\right)\cos\left(325\right)}$$
- step3: Calculate the trigonometric value:
  $$-445.080726$$
Let's break down the problem step by step.

### Part 1: Simplifying the First Expression

We need to evaluate the expression:

$$
\frac{3 \cos^4(90^\circ) - \sin^4(-10^\circ)}{\cos(55^\circ) \cos(325^\circ)}
$$

1. **Calculate $$\cos(90^\circ)$$**:
   - $$\cos(90^\circ) = 0$$
   - Therefore, $$\cos^4(90^\circ) = 0^4 = 0$$.

2. **Calculate $$\sin(-10^\circ)$$**:
   - $$\sin(-10^\circ) = -\sin(10^\circ)$$.
   - Thus, $$\sin^4(-10^\circ) = (-\sin(10^\circ))^4 = \sin^4(10^\circ)$$.

3. **Substituting back into the expression**:
   - The expression simplifies to:
   $$
   \frac{3 \cdot 0 - \sin^4(10^\circ)}{\cos(55^\circ) \cos(325^\circ)} = \frac{-\sin^4(10^\circ)}{\cos(55^\circ) \cos(325^\circ)}
   $$

4. **Calculate $$\cos(325^\circ)$$**:
   - $$\cos(325^\circ) = \cos(360^\circ - 35^\circ) = \cos(35^\circ)$$.

5. **Final Calculation**:
   - The final value of the expression is approximately $$-445.080726$$.

### Part 2: Solving the Second Equation

We need to solve the equation:

$$
\frac{\cos(40^\circ - x) \cos(x) - \sin(40^\circ - x) \sin(x)}{\sin(2) \cos(155^\circ)} = \pi
$$

1. **Recognize the left-hand side**:
   - The left-hand side can be simplified using the cosine of a sum:
   $$
   \cos(40^\circ - x + x) = \cos(40^\circ)
   $$
   - Thus, the equation simplifies to:
   $$
   \frac{\cos(40^\circ)}{\sin(2) \cos(155^\circ)} = \pi
   $$

2. **Check for solutions**:
   - The equation does not yield any real solutions, as indicated by the result $$x \in \varnothing$$.

### Summary of Results

- The value of the first expression is approximately $$-445.080726$$.
- The second equation has no real solutions ($$x \in \varnothing$$).
The first expression evaluates to approximately $$-445.080726$$, and the second equation has no real solutions.

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