If `10sin^4 alpha +15cos^4alpha=6` then the value of `9cosec^4 alpha + 8 sec^4 alpha-75` is

Answer

To solve the problem, we need to find the value of \( 9 \csc^4 \alpha + 8 \sec^4 \alpha - 75 \) given that \( 10 \sin^4 \alpha + 15 \cos^4 \alpha = 6 \).

### Step 1: Rewrite the given equation
We start with the equation:
\[ 10 \sin^4 \alpha + 15 \cos^4 \alpha = 6 \]
We can rewrite this as:
\[ 10 \sin^4 \alpha + 15 (1 - \sin^2 \alpha)^2 = 6 \]
Let \( x = \sin^2 \alpha \). Then, \( \cos^2 \alpha = 1 - x \), and we can express \( \cos^4 \alpha \) as \( (1 - x)^2 \).

### Step 2: Substitute and simplify
Substituting \( \cos^4 \alpha \):
\[ 10 x^2 + 15 (1 - x)^2 = 6 \]
Expanding the equation:
\[ 10 x^2 + 15 (1 - 2x + x^2) = 6 \]
\[ 10 x^2 + 15 - 30x + 15 x^2 = 6 \]
Combining like terms:
\[ 25 x^2 - 30x + 15 - 6 = 0 \]
\[ 25 x^2 - 30x + 9 = 0 \]

### Step 3: Solve the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 25, b = -30, c = 9 \):
\[ x = \frac{30 \pm \sqrt{(-30)^2 - 4 \cdot 25 \cdot 9}}{2 \cdot 25} \]
Calculating the discriminant:
\[ x = \frac{30 \pm \sqrt{900 - 900}}{50} = \frac{30 \pm 0}{50} = \frac{30}{50} = \frac{3}{5} \]
Thus, \( \sin^2 \alpha = \frac{3}{5} \) and consequently \( \cos^2 \alpha = 1 - \frac{3}{5} = \frac{2}{5} \).

### Step 4: Calculate \( \csc^4 \alpha \) and \( \sec^4 \alpha \)
Now we can find \( \csc^2 \alpha \) and \( \sec^2 \alpha \):
\[ \csc^2 \alpha = \frac{1}{\sin^2 \alpha} = \frac{1}{\frac{3}{5}} = \frac{5}{3} \]
\[ \sec^2 \alpha = \frac{1}{\cos^2 \alpha} = \frac{1}{\frac{2}{5}} = \frac{5}{2} \]
Now, we calculate \( \csc^4 \alpha \) and \( \sec^4 \alpha \):
\[ \csc^4 \alpha = \left(\frac{5}{3}\right)^4 = \frac{625}{81} \]
\[ \sec^4 \alpha = \left(\frac{5}{2}\right)^4 = \frac{625}{16} \]

### Step 5: Substitute into the expression
Now substitute these values into the expression:
\[ 9 \csc^4 \alpha + 8 \sec^4 \alpha - 75 = 9 \cdot \frac{625}{81} + 8 \cdot \frac{625}{16} - 75 \]
Calculating each term:
\[ 9 \cdot \frac{625}{81} = \frac{5625}{81} \]
\[ 8 \cdot \frac{625}{16} = \frac{5000}{16} \]
To add these fractions, we need a common denominator. The least common multiple of 81 and 16 is 1296.
Converting both fractions:
\[ \frac{5625}{81} = \frac{5625 \times 16}{1296} = \frac{90000}{1296} \]
\[ \frac{5000}{16} = \frac{5000 \times 81}{1296} = \frac{405000}{1296} \]
Adding these:
\[ \frac{90000 + 405000}{1296} = \frac{495000}{1296} \]
Now subtract 75:
\[ 75 = \frac{75 \times 1296}{1296} = \frac{97200}{1296} \]
Thus:
\[ \frac{495000 - 97200}{1296} = \frac{397800}{1296} \]
Simplifying:
\[ \frac{397800 \div 1296}{1296 \div 1296} = \frac{307}{1} = 307 \]

### Final Answer
The value of \( 9 \csc^4 \alpha + 8 \sec^4 \alpha - 75 \) is:
\[ \boxed{307} \]