What is the value of `(1)/( sin^(4) (90 - theta)) + (1)/([cos ^(2) (90 - theta)] - 1)` ?

Options

A. `tan 2 theta sec 2 theta `
B. `sec 4 theta `
C. `tan 4 theta `
D. `tan ^(2) theta sec ^(2) theta`

Answer

To solve the expression \( \frac{1}{\sin^4(90^\circ - \theta)} + \frac{1}{\cos^2(90^\circ - \theta) - 1} \), we can simplify it step by step.

### Step 1: Simplify the Trigonometric Functions
Using the co-function identities:
- \( \sin(90^\circ - \theta) = \cos(\theta) \)
- \( \cos(90^\circ - \theta) = \sin(\theta) \)

We can rewrite the expression:
\[ \frac{1}{\sin^4(90^\circ - \theta)} = \frac{1}{\cos^4(\theta)} \]
\[ \cos^2(90^\circ - \theta) - 1 = \sin^2(\theta) - 1 = -\cos^2(\theta) \]

### Step 2: Substitute into the Expression
Now substitute these identities into the original expression:
\[ \frac{1}{\cos^4(\theta)} + \frac{1}{-\cos^2(\theta)} \]

### Step 3: Combine the Terms
The second term can be rewritten as:
\[ \frac{1}{-\cos^2(\theta)} = -\frac{1}{\cos^2(\theta)} \]
Thus, the expression becomes:
\[ \frac{1}{\cos^4(\theta)} - \frac{1}{\cos^2(\theta)} \]

### Step 4: Find a Common Denominator
To combine these fractions, we need a common denominator, which is \( \cos^4(\theta) \):
\[ \frac{1}{\cos^4(\theta)} - \frac{\cos^2(\theta)}{\cos^4(\theta)} = \frac{1 - \cos^2(\theta)}{\cos^4(\theta)} \]

### Step 5: Simplify the Numerator
Using the identity \( 1 - \cos^2(\theta) = \sin^2(\theta) \), we can rewrite the expression:
\[ \frac{\sin^2(\theta)}{\cos^4(\theta)} \]

### Final Expression
The final simplified expression is:
\[ \frac{\sin^2(\theta)}{\cos^4(\theta)} \]

### Conclusion
Thus, the value of the original expression is:
\[ \frac{\sin^2(\theta)}{\cos^4(\theta)} \]