If `x in [0,2pi]` then the number of solution of the equation `81^(sin^2x)+81^(cos^2x)=30`

Answer

To solve the equation \( 81^{\sin^2 x} + 81^{\cos^2 x} = 30 \) for \( x \in [0, 2\pi] \), we can follow these steps:

### Step 1: Rewrite the Equation
We know that \( \sin^2 x + \cos^2 x = 1 \). Therefore, we can rewrite the equation as:
\[ 81^{\sin^2 x} + 81^{1 - \sin^2 x} = 30 \]

### Step 2: Substitute
Let \( y = 81^{\sin^2 x} \). Then, we can express \( 81^{\cos^2 x} \) as \( \frac{81}{y} \) since \( \cos^2 x = 1 - \sin^2 x \). The equation now becomes:
\[ y + \frac{81}{y} = 30 \]

### Step 3: Multiply through by \( y \)
To eliminate the fraction, multiply the entire equation by \( y \):
\[ y^2 + 81 = 30y \]

### Step 4: Rearrange the Equation
Rearranging gives us a standard quadratic equation:
\[ y^2 - 30y + 81 = 0 \]

### Step 5: Solve the Quadratic Equation
We can use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1 \), \( b = -30 \), and \( c = 81 \).
\[ y = \frac{30 \pm \sqrt{(-30)^2 - 4 \cdot 1 \cdot 81}}{2 \cdot 1} \]
\[ y = \frac{30 \pm \sqrt{900 - 324}}{2} \]
\[ y = \frac{30 \pm \sqrt{576}}{2} \]
\[ y = \frac{30 \pm 24}{2} \]

Calculating the two possible values for \( y \):
1. \( y = \frac{54}{2} = 27 \)
2. \( y = \frac{6}{2} = 3 \)

### Step 6: Back Substitute for \( \sin^2 x \)
Recall that \( y = 81^{\sin^2 x} \):
1. For \( y = 27 \):
\[ 81^{\sin^2 x} = 27 \implies \sin^2 x = \frac{1}{4} \implies \sin x = \pm \frac{1}{2} \]
- Solutions: \( x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} \)

2. For \( y = 3 \):
\[ 81^{\sin^2 x} = 3 \implies \sin^2 x = \frac{1}{4} \implies \sin x = \pm \frac{\sqrt{3}}{2} \]
- Solutions: \( x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3} \)

### Step 7: Count the Solutions
From both cases, we have:
- From \( y = 27 \): 4 solutions
- From \( y = 3 \): 4 solutions

### Total Solutions
Adding them together gives us a total of \( 4 + 4 = 8 \) solutions.

### Final Answer
The number of solutions of the equation \( 81^{\sin^2 x} + 81^{\cos^2 x} = 30 \) in the interval \( [0, 2\pi] \) is \( \boxed{8} \).

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