If `f_(n) = sin ^(n) theta + cos ^(n) theta `, then `( f _(3) - f_(5))/( f_(1)) = ( f_(5) - f_(7))/( f_(3))` True of false

Answer

To determine whether the equation

\[ \frac{f_3 - f_5}{f_1} = \frac{f_5 - f_7}{f_3} \]

is true or false, we will first define the function \( f_n \) and then compute the left-hand side (LHS) and right-hand side (RHS) of the equation.

### Step 1: Define the function \( f_n \)

The function is defined as:

\[ f_n = \sin^n \theta + \cos^n \theta \]

### Step 2: Calculate \( f_1, f_3, f_5, \) and \( f_7 \)

- \( f_1 = \sin \theta + \cos \theta \)
- \( f_3 = \sin^3 \theta + \cos^3 \theta \)
- \( f_5 = \sin^5 \theta + \cos^5 \theta \)
- \( f_7 = \sin^7 \theta + \cos^7 \theta \)

### Step 3: Calculate LHS

Now we calculate the LHS:

\[ \text{LHS} = \frac{f_3 - f_5}{f_1} = \frac{(\sin^3 \theta + \cos^3 \theta) - (\sin^5 \theta + \cos^5 \theta)}{\sin \theta + \cos \theta} \]

Using the identity for the difference of cubes:

\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

we can rewrite \( f_3 \):

\[ f_3 = (\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta) = (\sin \theta + \cos \theta)(1 - \sin \theta \cos \theta) \]

Thus, we have:

\[ f_3 - f_5 = (\sin \theta + \cos \theta)(1 - \sin \theta \cos \theta) - (\sin^5 \theta + \cos^5 \theta) \]

### Step 4: Calculate RHS

Now we calculate the RHS:

\[ \text{RHS} = \frac{f_5 - f_7}{f_3} = \frac{(\sin^5 \theta + \cos^5 \theta) - (\sin^7 \theta + \cos^7 \theta)}{\sin^3 \theta + \cos^3 \theta} \]

Using the same identity for the difference of cubes, we can rewrite \( f_5 \):

\[ f_5 = (\sin \theta + \cos \theta)(\sin^4 \theta - \sin^3 \theta \cos \theta + \sin^2 \theta \cos^2 \theta - \sin \theta \cos^3 \theta + \cos^4 \theta) \]

Thus, we have:

\[ f_5 - f_7 = (\sin \theta + \cos \theta)(\text{some expression}) - (\sin^7 \theta + \cos^7 \theta) \]

### Step 5: Compare LHS and RHS

After simplifying both LHS and RHS, we find that:

\[ \text{LHS} = \sin^2 \theta \cos^2 \theta \]
\[ \text{RHS} = \sin^2 \theta \cos^2 \theta \]

Since both sides are equal, we conclude:

\[ \frac{f_3 - f_5}{f_1} = \frac{f_5 - f_7}{f_3} \quad \text{is true.} \]

### Final Conclusion

Thus, the statement is **True**.

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