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

Options

A. 150
B. 175
C. 225
D. 250

Explanation

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

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

We want to find the value of:
\[ 8 \csc^6 \alpha + 27 \sec^6 \alpha \]

### Step 1: Rewrite the equation in terms of \(\cos^4 \alpha\)

Divide the entire equation by \(\cos^4 \alpha\):
\[ \frac{15 \sin^4 \alpha}{\cos^4 \alpha} + 10 = \frac{6}{\cos^4 \alpha} \]

Using the identity \(\sin^2 \alpha = 1 - \cos^2 \alpha\), we can express \(\sin^4 \alpha\) as:
\[ \sin^4 \alpha = (1 - \cos^2 \alpha)^2 = 1 - 2\cos^2 \alpha + \cos^4 \alpha \]
Thus, we have:
\[ \frac{15(1 - 2\cos^2 \alpha + \cos^4 \alpha)}{\cos^4 \alpha} + 10 = \frac{6}{\cos^4 \alpha} \]

### Step 2: Simplify the equation

This simplifies to:
\[ 15 \left(\frac{1}{\cos^4 \alpha} - \frac{2}{\cos^2 \alpha} + 1\right) + 10 = \frac{6}{\cos^4 \alpha} \]
Multiplying through by \(\cos^4 \alpha\) to eliminate the denominators:
\[ 15(1 - 2\cos^2 \alpha + \cos^4 \alpha) + 10\cos^4 \alpha = 6 \]
This gives:
\[ 15 - 30\cos^2 \alpha + 25\cos^4 \alpha = 6 \]

### Step 3: Rearranging the equation

Rearranging yields:
\[ 25\cos^4 \alpha - 30\cos^2 \alpha + 9 = 0 \]

Let \(x = \cos^2 \alpha\). The equation becomes:
\[ 25x^2 - 30x + 9 = 0 \]

### Step 4: Solve the quadratic equation

Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 25\), \(b = -30\), and \(c = 9\):
\[ x = \frac{30 \pm \sqrt{(-30)^2 - 4 \cdot 25 \cdot 9}}{2 \cdot 25} \]
\[ x = \frac{30 \pm \sqrt{900 - 900}}{50} = \frac{30}{50} = \frac{3}{5} \]

Thus,
\[ \cos^2 \alpha = \frac{3}{5} \]

### Step 5: Find \(\sin^2 \alpha\)

Using \(\sin^2 \alpha + \cos^2 \alpha = 1\):
\[ \sin^2 \alpha = 1 - \cos^2 \alpha = 1 - \frac{3}{5} = \frac{2}{5} \]

### Step 6: Calculate \(\csc^6 \alpha\) and \(\sec^6 \alpha\)

Now we can calculate:
\[ \csc^2 \alpha = \frac{1}{\sin^2 \alpha} = \frac{5}{2} \quad \Rightarrow \quad \csc^6 \alpha = \left(\frac{5}{2}\right)^3 = \frac{125}{8} \]
\[ \sec^2 \alpha = \frac{1}{\cos^2 \alpha} = \frac{5}{3} \quad \Rightarrow \quad \sec^6 \alpha = \left(\frac{5}{3}\right)^3 = \frac{125}{27} \]

### Step 7: Substitute into the expression

Now substituting into the expression:
\[ 8 \csc^6 \alpha + 27 \sec^6 \alpha = 8 \cdot \frac{125}{8} + 27 \cdot \frac{125}{27} \]
\[ = 125 + 125 = 250 \]

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