m=cosec x-sinx

n=secx-cosx

find value of f(m)

m^2n^2(m^2+n^2+5)=?

To solve the problem where we have the expressions \( m = \csc x - \sin x \) and \( n = \sec x - \cos x \), and we need to find the value of \( f(m) = m^2 n^2 (m^2 + n^2 + 5) \), let's break this down step by step.

Step 1: Simplifying m and n

First, we need to simplify the expressions for \( m \) and \( n \).

• For \( m = \csc x - \sin x \):

We know that \( \csc x = \frac{1}{\sin x} \). Therefore, we can rewrite \( m \) as:

\( m = \frac{1}{\sin x} - \sin x = \frac{1 - \sin^2 x}{\sin x} = \frac{\cos^2 x}{\sin x} \)

• For \( n = \sec x - \cos x \):

Similarly, since \( \sec x = \frac{1}{\cos x} \), we have:

\( n = \frac{1}{\cos x} - \cos x = \frac{1 - \cos^2 x}{\cos x} = \frac{\sin^2 x}{\cos x} \)

Step 2: Finding m^2 and n^2

Next, we need to calculate \( m^2 \) and \( n^2 \).

• Calculating \( m^2 \):

We find:

\( m^2 = \left( \frac{\cos^2 x}{\sin x} \right)^2 = \frac{\cos^4 x}{\sin^2 x} \)

• Calculating \( n^2 \):

For \( n \), we have:

\( n^2 = \left( \frac{\sin^2 x}{\cos x} \right)^2 = \frac{\sin^4 x}{\cos^2 x} \)

Step 3: Finding m^2 n^2

The next step is to calculate \( m^2 n^2 \):

\( m^2 n^2 = \frac{\cos^4 x}{\sin^2 x} \cdot \frac{\sin^4 x}{\cos^2 x} = \frac{\cos^2 x \sin^4 x}{\sin^2 x} = \frac{\cos^2 x \sin^2 x}{1} = \cos^2 x \sin^2 x \)

Step 4: Evaluating m^2 + n^2

Now, we move on to find \( m^2 + n^2 \):

\( m^2 + n^2 = \frac{\cos^4 x}{\sin^2 x} + \frac{\sin^4 x}{\cos^2 x} \)

This can be simplified using a common denominator:

\( = \frac{\cos^6 x + \sin^6 x}{\sin^2 x \cos^2 x} \)

Step 5: Final Calculation

Finally, we need to evaluate the expression \( f(m) = m^2 n^2 (m^2 + n^2 + 5) \):

Substituting our earlier findings:

\( f(m) = \cos^2 x \sin^2 x \left( \frac{\cos^6 x + \sin^6 x}{\sin^2 x \cos^2 x} + 5 \right) \)

This simplifies to:

\( = \cos^2 x \sin^2 x \left( \frac{\cos^6 x + \sin^6 x + 5 \sin^2 x \cos^2 x}{\sin^2 x \cos^2 x} \right) \)

Thus, the final expression will give us the required value. Depending on the specific angle \( x \) you choose, you can substitute it to find a numerical result.

In conclusion, the process involves simplifying trigonometric identities, then calculating the necessary squares and combining them effectively to arrive at the final expression. This approach reinforces the importance of understanding trigonometric identities in solving complex expressions!