The value of \(\rm \left[\frac{\cos 50^\circ}{1+\sin 50^\circ}\right]+\left[\frac{1+\sin 50^\circ}{\cos50^\circ}\right]\) is:

Given:

We need to find the value of: \(\rm \left[\frac{\cos 50^\circ}{1+\sin 50^\circ}\right]+\left[\frac{1+\sin 50^\circ}{\cos50^\circ}\right]\)

Formula Used:

cos² θ + sin² θ = 1

Calculation:

\(\rm \left[\frac{\cos 50^\circ}{1+\sin 50^\circ}\right]+\left[\frac{1+\sin 50^\circ}{\cos50^\circ}\right]\)

Take the LCM of the two terms:

\(\frac{cos² 50° + (1 + sin 50°)² }{(1 + sin 50°) × cos 50°}\)

Expanding the numerator:

cos² 50° + (1 + 2sin 50° + sin² 50°)

Using the identity cos² θ + sin² θ = 1:

1 + (1 + 2sin 50°)

2 + 2sin 50°

Now, the expression becomes:

\(\frac{2 + 2sin 50°}{(1 + sin 50°) × cos 50°}\)

Factor out 2 from the numerator:

\(\frac{2(1 + sin 50°)}{(1 + sin 50°) × cos 50°}\)

Cancel out the common term (1 + sin 50°):

2 / cos 50°

The value of 1 / cos 50° is sec 50°.

Therefore, the expression simplifies to: 2 sec 50°

∴ The value of the given expression is 2 sec 50°.