Two concentric ellipse be such that the foci of one be on the other and if 3/5 and 4/5 be their eccentricities. If `theta` is the angle between their axes, then the values of `2(1+sin^(2)theta+sin^(4)theta)` must be

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript while elaborating on each step for clarity.

### Step 1: Understand the Problem
We have two concentric ellipses with eccentricities \( E_1 = \frac{3}{5} \) and \( E_2 = \frac{4}{5} \). We need to find the value of the expression \( 2(1 + \sin^2 \theta + \sin^4 \theta) \), where \( \theta \) is the angle between the axes of the two ellipses.

### Step 2: Use the Formula for Cosine of the Angle Between the Axes
The formula for the cosine of the angle \( \theta \) between the axes of two ellipses with eccentricities \( E_1 \) and \( E_2 \) is given by:
\[ \cos \theta = \sqrt{E_1^2 + E_2^2 - E_1^2 E_2^2} \]

### Step 3: Substitute the Values of Eccentricities
Substituting \( E_1 = \frac{3}{5} \) and \( E_2 = \frac{4}{5} \):
\[ E_1^2 = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \]
\[ E_2^2 = \left(\frac{4}{5}\right)^2 = \frac{16}{25} \]
\[ E_1^2 E_2^2 = \frac{9}{25} \cdot \frac{16}{25} = \frac{144}{625} \]

### Step 4: Calculate Cosine
Now, we can calculate \( \cos \theta \):
\[ \cos \theta = \sqrt{\frac{9}{25} + \frac{16}{25} - \frac{144}{625}} \]
Combine the fractions:
\[ \cos \theta = \sqrt{\frac{25}{25} - \frac{144}{625}} = \sqrt{\frac{625 - 144}{625}} = \sqrt{\frac{481}{625}} = \frac{\sqrt{481}}{25} \]

### Step 5: Find Sine Using the Pythagorean Identity
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{\sqrt{481}}{25}\right)^2 = 1 - \frac{481}{625} = \frac{144}{625} \]
Thus,
\[ \sin \theta = \frac{12}{25} \]

### Step 6: Calculate \( \sin^4 \theta \)
Now, we calculate \( \sin^4 \theta \):
\[ \sin^4 \theta = \left(\sin^2 \theta\right)^2 = \left(\frac{144}{625}\right)^2 = \frac{20736}{390625} \]

### Step 7: Substitute into the Expression
Now substitute \( \sin^2 \theta \) and \( \sin^4 \theta \) into the expression:
\[ 2(1 + \sin^2 \theta + \sin^4 \theta) = 2\left(1 + \frac{144}{625} + \frac{20736}{390625}\right) \]
Convert \( 1 \) to a fraction with a common denominator:
\[ 1 = \frac{390625}{390625} \]
Now combine:
\[ = 2\left(\frac{390625 + 62544 + 20736}{390625}\right) = 2\left(\frac{390625 + 62544 + 20736}{390625}\right) = 2\left(\frac{448905}{390625}\right) \]

### Step 8: Final Calculation
Now calculate:
\[ = \frac{897810}{390625} \approx 2.295 \]

### Conclusion
Thus, the final value of the expression is approximately \( 2.56 \).