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

Explanation

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

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

### Step 1: Rewrite the equation in terms of \(\sin^2\alpha\) and \(\cos^2\alpha\)

Let \( x = \sin^2\alpha \) and \( y = \cos^2\alpha \). We know that \( x + y = 1 \). Therefore, we can express \(\cos^4\alpha\) as:

\[ \cos^4\alpha = (1 - \sin^2\alpha)^2 = (1 - x)^2 = 1 - 2x + x^2 \]

Substituting this into the original equation gives:

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

### Step 2: Simplify the equation

Expanding the equation:

\[ 15x^2 + 10 - 20x + 10x^2 = 6 \]

Combine like terms:

\[ (15x^2 + 10x^2) - 20x + 10 = 6 \]

This simplifies to:

\[ 25x^2 - 20x + 10 = 6 \]

### Step 3: Rearranging the equation

Now, we can rearrange the equation:

\[ 25x^2 - 20x + 4 = 0 \]

### Step 4: Solve the quadratic equation

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 25 \), \( b = -20 \), and \( c = 4 \).

Calculating the discriminant:

\[ b^2 - 4ac = (-20)^2 - 4 \cdot 25 \cdot 4 = 400 - 400 = 0 \]

Since the discriminant is zero, there is one real solution:

\[ x = \frac{-(-20) \pm \sqrt{0}}{2 \cdot 25} = \frac{20}{50} = \frac{2}{5} \]

Thus,

\[ \sin^2\alpha = \frac{2}{5} \]

And since \( \cos^2\alpha = 1 - \sin^2\alpha \):

\[ \cos^2\alpha = 1 - \frac{2}{5} = \frac{3}{5} \]

### Step 5: Find \( \csc^2\alpha \) and \( \sec^2\alpha \)

Now, we can find:

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

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

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

Now we calculate:

\[ \csc^6\alpha = \left(\frac{5}{2}\right)^6 = \frac{15625}{64} \]

\[ \sec^6\alpha = \left(\frac{5}{3}\right)^6 = \frac{15625}{729} \]

Now substituting back into the expression:

\[ 8\csc^6\alpha = 8 \cdot \frac{15625}{64} = \frac{125000}{64} \]

\[ 27\sec^6\alpha = 27 \cdot \frac{15625}{729} = \frac{421875}{729} \]

### Step 7: Find a common denominator and add

The common denominator of \( 64 \) and \( 729 \) is \( 46656 \):

Convert \( \frac{125000}{64} \) to have a denominator of \( 46656 \):

\[ \frac{125000 \cdot 729}{46656} = \frac{91250000}{46656} \]

Convert \( \frac{421875}{729} \) to have a denominator of \( 46656 \):

\[ \frac{421875 \cdot 64}{46656} = \frac{26925000}{46656} \]

Now add them:

\[ \frac{91250000 + 26925000}{46656} = \frac{118375000}{46656} \]

Thus, the final answer is:

\[ 8\csc^6\alpha + 27\sec^6\alpha = \frac{118375000}{46656} \]