If `15sin^4alpha+10cos^4alpha=6 ` , then `27sec^6alpha+8cosec^6 alpha =`

Answer

To solve the equation \( 15\sin^4\alpha + 10\cos^4\alpha = 6 \) and find the value of \( 27\sec^6\alpha + 8\csc^6\alpha \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 15\sin^4\alpha + 10\cos^4\alpha = 6 \]

### Step 2: Substitute \( x = \sin^2\alpha \) and \( y = \cos^2\alpha \)
Using the identity \( \sin^2\alpha + \cos^2\alpha = 1 \), we can express \( \cos^2\alpha \) as \( y = 1 - x \). Thus, we can rewrite the equation as:
\[ 15x^2 + 10(1 - x)^2 = 6 \]

### Step 3: Expand and simplify
Now, expand the equation:
\[ 15x^2 + 10(1 - 2x + x^2) = 6 \]
\[ 15x^2 + 10 - 20x + 10x^2 = 6 \]
Combine like terms:
\[ 25x^2 - 20x + 10 = 6 \]
Subtract 6 from both sides:
\[ 25x^2 - 20x + 4 = 0 \]

### Step 4: Solve the quadratic equation
Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 25 \), \( b = -20 \), and \( c = 4 \).
\[ x = \frac{20 \pm \sqrt{(-20)^2 - 4 \cdot 25 \cdot 4}}{2 \cdot 25} \]
\[ x = \frac{20 \pm \sqrt{400 - 400}}{50} \]
\[ x = \frac{20}{50} = \frac{2}{5} \]

### Step 5: Find \( \sin^2\alpha \) and \( \cos^2\alpha \)
Since \( x = \sin^2\alpha \):
\[ \sin^2\alpha = \frac{2}{5} \]
Then, using \( \cos^2\alpha = 1 - \sin^2\alpha \):
\[ \cos^2\alpha = 1 - \frac{2}{5} = \frac{3}{5} \]

### Step 6: Calculate \( \sec^2\alpha \) and \( \csc^2\alpha \)
Now we can find \( \sec^2\alpha \) and \( \csc^2\alpha \):
\[ \sec^2\alpha = \frac{1}{\cos^2\alpha} = \frac{1}{\frac{3}{5}} = \frac{5}{3} \]
\[ \csc^2\alpha = \frac{1}{\sin^2\alpha} = \frac{1}{\frac{2}{5}} = \frac{5}{2} \]

### Step 7: Calculate \( \sec^6\alpha \) and \( \csc^6\alpha \)
Now we find \( \sec^6\alpha \) and \( \csc^6\alpha \):
\[ \sec^6\alpha = \left(\frac{5}{3}\right)^3 = \frac{125}{27} \]
\[ \csc^6\alpha = \left(\frac{5}{2}\right)^3 = \frac{125}{8} \]

### Step 8: Substitute into the expression
Now substitute these values into the expression \( 27\sec^6\alpha + 8\csc^6\alpha \):
\[ 27\sec^6\alpha + 8\csc^6\alpha = 27 \cdot \frac{125}{27} + 8 \cdot \frac{125}{8} \]
\[ = 125 + 125 = 250 \]

### Final Answer
Thus, the value of \( 27\sec^6\alpha + 8\csc^6\alpha \) is:
\[ \boxed{250} \]