tan $$ (36-(x) an (48-x)=-2 $$

Question

tan $$ (36-(x) 	an (48-x)=-2 $$

UpStudy AI · Accepted Answer

Let's solve the equation:

$$
\tan(36^\circ - x) \cdot \tan(48^\circ - x) = -2
$$

### Step 1: Simplify the Equation

Let’s denote:
$$
A = 36^\circ - x \quad \text{and} \quad B = 48^\circ - x
$$
So the equation becomes:
$$
\tan A \cdot \tan B = -2
$$

### Step 2: Use Trigonometric Identities

Recall the trigonometric identity:
$$
\tan A \cdot \tan B = \frac{\cos(A - B) - \cos(A + B)}{2 \cos A \cos B}
$$
However, applying this identity directly leads to a rather complex equation. Instead, let's approach the problem by manipulating the equation to isolate $$ x $$.

### Step 3: Rearrange the Equation

Starting with:
$$
\tan(36^\circ - x) \cdot \tan(48^\circ - x) = -2
$$

Let’s substitute $$ y = x $$ for simplicity:
$$
\tan(36^\circ - y) \cdot \tan(48^\circ - y) = -2
$$

### Step 4: Analyze Possible Solutions

Attempting to solve this equation numerically reveals that there are **no real solutions**. Here's why:

1. **Range of Tangent Function**: The tangent function, $$ \tan(\theta) $$, ranges from $$-\infty$$ to $$+\infty$$. However, when considering the product $$ \tan(A) \cdot \tan(B) $$, achieving a specific constant (like $$-2$$) depends on the angles $$ A $$ and $$ B $$.

2. **Identity Constraints**: By manipulating the equation using trigonometric identities, we derive:
   $$
   \cos(84^\circ - 2x) = -3 \cos(12^\circ)
   $$
   Evaluating the right side:
   $$
   -3 \cos(12^\circ) \approx -3 \times 0.9781 \approx -2.9343
   $$
   However, the cosine of any real angle must lie within $$[-1, 1]$$. Since $$-2.9343$$ is outside this range, the equation has no real solutions.

### Conclusion

There are **no real values of $$ x $$** that satisfy the equation:
$$
\tan(36^\circ - x) \cdot \tan(48^\circ - x) = -2
$$

## Final Answer

There are no real numbers x that satisfy the equation tan (36 – x) × tan (48 – x) = –2.
There are no real solutions for $$ x $$ in the equation $$ \tan(36^\circ - x) \cdot \tan(48^\circ - x) = -2 $$.

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