Total number of solutions of `16^(sin^(2)x)+16^(cos^(2)x)=10" in "[0,2pi]` are

Options

A. 4
B. 8
C. 12
D. 16

Answer

To find the total number of solutions for the equation \( 16^{\sin^2 x} + 16^{\cos^2 x} = 10 \) in the interval \([0, 2\pi]\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 16^{\sin^2 x} + 16^{\cos^2 x} = 10 \]
Using the identity \( \cos^2 x = 1 - \sin^2 x \), we can rewrite \( 16^{\cos^2 x} \) as \( 16^{1 - \sin^2 x} \):
\[ 16^{\sin^2 x} + 16^{1 - \sin^2 x} = 10 \]

### Step 2: Simplify the equation
We can express \( 16^{1 - \sin^2 x} \) as \( \frac{16}{16^{\sin^2 x}} \):
\[ 16^{\sin^2 x} + \frac{16}{16^{\sin^2 x}} = 10 \]
Let \( y = 16^{\sin^2 x} \). Then the equation becomes:
\[ y + \frac{16}{y} = 10 \]

### Step 3: Multiply through by \( y \)
To eliminate the fraction, multiply both sides by \( y \):
\[ y^2 + 16 = 10y \]
Rearranging gives us a quadratic equation:
\[ y^2 - 10y + 16 = 0 \]

### Step 4: Solve the quadratic equation
We can use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -10, c = 16 \):
\[ y = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 16}}{2 \cdot 1} \]
Calculating the discriminant:
\[ = \frac{10 \pm \sqrt{100 - 64}}{2} = \frac{10 \pm \sqrt{36}}{2} = \frac{10 \pm 6}{2} \]
This gives us two solutions:
\[ y_1 = \frac{16}{2} = 8 \quad \text{and} \quad y_2 = \frac{4}{2} = 2 \]

### Step 5: Find \( \sin^2 x \) for each \( y \)
1. For \( y = 8 \):
\[ 16^{\sin^2 x} = 8 \implies \sin^2 x = \frac{3}{4} \implies \sin x = \pm \frac{\sqrt{3}}{2} \]
The angles corresponding to \( \sin x = \frac{\sqrt{3}}{2} \) are:
\[ x = \frac{\pi}{3}, \quad \frac{2\pi}{3} \]
The angles corresponding to \( \sin x = -\frac{\sqrt{3}}{2} \) are:
\[ x = \frac{4\pi}{3}, \quad \frac{5\pi}{3} \]
Total solutions from \( y = 8 \): **4 solutions**.

2. For \( y = 2 \):
\[ 16^{\sin^2 x} = 2 \implies \sin^2 x = \frac{1}{4} \implies \sin x = \pm \frac{1}{2} \]
The angles corresponding to \( \sin x = \frac{1}{2} \) are:
\[ x = \frac{\pi}{6}, \quad \frac{5\pi}{6} \]
The angles corresponding to \( \sin x = -\frac{1}{2} \) are:
\[ x = \frac{7\pi}{6}, \quad \frac{11\pi}{6} \]
Total solutions from \( y = 2 \): **4 solutions**.

### Step 6: Calculate the total number of solutions
Adding the solutions from both cases:
\[ 4 + 4 = 8 \]

### Final Answer
The total number of solutions of the equation \( 16^{\sin^2 x} + 16^{\cos^2 x} = 10 \) in the interval \([0, 2\pi]\) is **8**.

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