Number of roots of the equation `2^(tan(x-pi/4))-2(0. 25)^sin^(3((x-pi/4))/(cos2x))+1=0,i s_______`

Explanation

To find the number of roots of the equation

\[ 2^{\tan(x - \frac{\pi}{4})} - 2(0.25)^{\frac{\sin^3(x - \frac{\pi}{4})}{\cos(2x)}} + 1 = 0, \]

we can follow these steps:

### Step 1: Rewrite the Equation
First, we can rewrite \(0.25\) as \(\left(\frac{1}{4}\right)\) or \(\left(\frac{1}{2}\right)^2\). Thus, we have:

\[ 0.25 = \left(\frac{1}{2}\right)^2 \implies (0.25)^{\frac{\sin^3(x - \frac{\pi}{4})}{\cos(2x)}} = \left(\frac{1}{2}\right)^{\frac{2\sin^3(x - \frac{\pi}{4})}{\cos(2x)}} \]

### Step 2: Substitute and Simplify
Substituting this back into the equation gives:

\[ 2^{\tan(x - \frac{\pi}{4})} - 2 \cdot \left(\frac{1}{2}\right)^{\frac{2\sin^3(x - \frac{\pi}{4})}{\cos(2x)}} + 1 = 0 \]

### Step 3: Analyze the Function
Let \(y = \tan(x - \frac{\pi}{4})\). The equation can be rewritten as:

\[ 2^y - 2 \cdot 2^{-\frac{2\sin^3(x - \frac{\pi}{4})}{\cos(2x)}} + 1 = 0 \]

### Step 4: Set Up for Roots
We need to analyze the behavior of the function \(f(y) = 2^y - 2 \cdot 2^{-\frac{2\sin^3(x - \frac{\pi}{4})}{\cos(2x)}} + 1\).

### Step 5: Find Critical Points
To find the roots, we can set \(f(y) = 0\) and analyze the critical points. We can also look for intersections of the two components of the equation.

### Step 6: Check for Validity of Roots
We need to check if the values of \(x\) that satisfy \(y = \tan(x - \frac{\pi}{4})\) lead to valid outputs for the original equation.

### Step 7: Evaluate at Specific Points
Evaluate at \(x = \frac{\pi}{4}\):
\[ \tan(0) = 0 \implies 2^0 - 2(0.25)^{0} + 1 = 0 - 2 + 1 = -1 \neq 0 \]

Thus, \(x = \frac{\pi}{4}\) does not satisfy the equation.

### Step 8: Conclusion
Since we have checked the critical point and found that it does not satisfy the equation, we conclude that there are no roots.

The total number of roots of the equation is:

\[ \boxed{0} \]

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