Show that `(1)/( 1 + cos (90^(@) - theta)) + (1)/(1 - cos (90^(@) - theta )) = 2 cosec ^(2) (90 ^(@) - theta )`

Answer

To show that

\[ \frac{1}{1 + \cos(90^\circ - \theta)} + \frac{1}{1 - \cos(90^\circ - \theta)} = 2 \csc^2(90^\circ - \theta) \]

we will follow these steps:

### Step 1: Use the co-function identity
We know that

\[ \cos(90^\circ - \theta) = \sin(\theta) \]

### Step 2: Substitute the identity into the equation
Substituting the identity into the left-hand side (LHS):

\[ \frac{1}{1 + \sin(\theta)} + \frac{1}{1 - \sin(\theta)} \]

### Step 3: Find a common denominator
The common denominator for the two fractions is

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

### Step 4: Rewrite the fractions
Now we rewrite the LHS with the common denominator:

\[ \frac{(1 - \sin(\theta)) + (1 + \sin(\theta))}{(1 + \sin(\theta))(1 - \sin(\theta))} \]

### Step 5: Simplify the numerator
The numerator simplifies to:

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

So, we have:

\[ \frac{2}{1 - \sin^2(\theta)} \]

### Step 6: Use the Pythagorean identity
Using the Pythagorean identity \(1 - \sin^2(\theta) = \cos^2(\theta)\), we can rewrite the expression:

\[ \frac{2}{\cos^2(\theta)} \]

### Step 7: Rewrite in terms of cosecant
We know that

\[ \frac{1}{\cos^2(\theta)} = \sec^2(\theta) \]

Thus, we can write:

\[ \frac{2}{\cos^2(\theta)} = 2 \sec^2(\theta) \]

### Step 8: Relate secant to cosecant
Using the identity \(\sec^2(\theta) = \csc^2(90^\circ - \theta)\), we can rewrite the expression as:

\[ 2 \sec^2(\theta) = 2 \csc^2(90^\circ - \theta) \]

### Conclusion
Thus, we have shown that:

\[ \frac{1}{1 + \cos(90^\circ - \theta)} + \frac{1}{1 - \cos(90^\circ - \theta)} = 2 \csc^2(90^\circ - \theta) \]