If `10sin^4 alpha+15 cos^4 alpha=6,` then find the value of `27 cosec^6 alpha+8sec^6 alpha.`

Options

A. 125
B. 250
C. 50
D. 75

Explanation

To solve the problem, we start with the equation given:

\[ 10 \sin^4 \alpha + 15 \cos^4 \alpha = 6 \]

### Step 1: Rewrite the equation using the identity
We know that \( \sin^2 \alpha + \cos^2 \alpha = 1 \). Therefore, we can express \( \sin^4 \alpha \) and \( \cos^4 \alpha \) in terms of \( \sin^2 \alpha \) and \( \cos^2 \alpha \):

\[ \sin^4 \alpha = (\sin^2 \alpha)^2 \quad \text{and} \quad \cos^4 \alpha = (\cos^2 \alpha)^2 \]

Let \( x = \sin^2 \alpha \) and \( y = \cos^2 \alpha \). Then, we have:

\[ x + y = 1 \]

Substituting \( y = 1 - x \) into the original equation gives:

\[ 10x^2 + 15(1 - x)^2 = 6 \]

### Step 2: Expand and simplify
Now, we expand the equation:

\[ 10x^2 + 15(1 - 2x + x^2) = 6 \]

This simplifies to:

\[ 10x^2 + 15 - 30x + 15x^2 = 6 \]

Combining like terms, we get:

\[ 25x^2 - 30x + 15 - 6 = 0 \]

Thus, the equation becomes:

\[ 25x^2 - 30x + 9 = 0 \]

### Step 3: Solve the quadratic equation
Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 25 \), \( b = -30 \), and \( c = 9 \).

Calculating the discriminant:

\[ b^2 - 4ac = (-30)^2 - 4 \cdot 25 \cdot 9 = 900 - 900 = 0 \]

Since the discriminant is zero, there is one repeated root:

\[ x = \frac{30}{2 \cdot 25} = \frac{30}{50} = \frac{3}{5} \]

### Step 4: Find \( \cos^2 \alpha \)
Using \( x + y = 1 \):

\[ \sin^2 \alpha = \frac{3}{5} \quad \text{and} \quad \cos^2 \alpha = 1 - \frac{3}{5} = \frac{2}{5} \]

### Step 5: Calculate \( 27 \csc^6 \alpha + 8 \sec^6 \alpha \)
We know:

\[ \csc^2 \alpha = \frac{1}{\sin^2 \alpha} = \frac{1}{\frac{3}{5}} = \frac{5}{3} \]
\[ \sec^2 \alpha = \frac{1}{\cos^2 \alpha} = \frac{1}{\frac{2}{5}} = \frac{5}{2} \]

Now, we calculate \( \csc^6 \alpha \) and \( \sec^6 \alpha \):

\[ \csc^6 \alpha = \left(\frac{5}{3}\right)^3 = \frac{125}{27} \]
\[ \sec^6 \alpha = \left(\frac{5}{2}\right)^3 = \frac{125}{8} \]

### Step 6: Substitute into the expression
Now substituting these into the expression:

\[ 27 \csc^6 \alpha + 8 \sec^6 \alpha = 27 \cdot \frac{125}{27} + 8 \cdot \frac{125}{8} \]

This simplifies to:

\[ 125 + 125 = 250 \]

### Final Answer
Thus, the value of \( 27 \csc^6 \alpha + 8 \sec^6 \alpha \) is:

\[ \boxed{250} \]