Find the value of
`(sectheta-costheta)^(2)+(cosectheta-sintheta)^(2)-(cottheta-tantheta)^(2)`.

Options

A. 0
B. `(1)/(2)`
C. 1
D. 2

Answer

To solve the expression \((\sec \theta - \cos \theta)^2 + (\csc \theta - \sin \theta)^2 - (\cot \theta - \tan \theta)^2\), we can break it down step by step.

### Step 1: Expand the first term \((\sec \theta - \cos \theta)^2\)

Using the identity \(\sec \theta = \frac{1}{\cos \theta}\), we have:

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

This can be simplified as:

\[ \left(\frac{1 - \cos^2 \theta}{\cos \theta}\right)^2 = \left(\frac{\sin^2 \theta}{\cos \theta}\right)^2 = \frac{\sin^4 \theta}{\cos^2 \theta} \]

### Step 2: Expand the second term \((\csc \theta - \sin \theta)^2\)

Using the identity \(\csc \theta = \frac{1}{\sin \theta}\), we have:

\[ (\csc \theta - \sin \theta)^2 = \left(\frac{1}{\sin \theta} - \sin \theta\right)^2 \]

This can be simplified as:

\[ \left(\frac{1 - \sin^2 \theta}{\sin \theta}\right)^2 = \left(\frac{\cos^2 \theta}{\sin \theta}\right)^2 = \frac{\cos^4 \theta}{\sin^2 \theta} \]

### Step 3: Expand the third term \((\cot \theta - \tan \theta)^2\)

Using the identities \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), we have:

\[ (\cot \theta - \tan \theta)^2 = \left(\frac{\cos \theta}{\sin \theta} - \frac{\sin \theta}{\cos \theta}\right)^2 \]

This can be simplified as:

\[ \left(\frac{\cos^2 \theta - \sin^2 \theta}{\sin \theta \cos \theta}\right)^2 = \frac{(\cos^2 \theta - \sin^2 \theta)^2}{\sin^2 \theta \cos^2 \theta} \]

### Step 4: Combine all the terms

Now we can combine all the terms:

\[ \frac{\sin^4 \theta}{\cos^2 \theta} + \frac{\cos^4 \theta}{\sin^2 \theta} - \frac{(\cos^2 \theta - \sin^2 \theta)^2}{\sin^2 \theta \cos^2 \theta} \]

### Step 5: Simplify the expression

To simplify this, we can find a common denominator, which is \(\sin^2 \theta \cos^2 \theta\):

\[ \frac{\sin^4 \theta \sin^2 \theta + \cos^4 \theta \cos^2 \theta - (\cos^2 \theta - \sin^2 \theta)^2}{\sin^2 \theta \cos^2 \theta} \]

### Step 6: Recognize the identity

Notice that \((\cos^2 \theta - \sin^2 \theta)^2 = \cos^4 \theta - 2\cos^2 \theta \sin^2 \theta + \sin^4 \theta\). Thus, the numerator simplifies to:

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

This results in:

\[ 4\cos^2 \theta \sin^2 \theta \]

### Final Step: Evaluate the expression

Thus, the expression simplifies to:

\[ \frac{4\cos^2 \theta \sin^2 \theta}{\sin^2 \theta \cos^2 \theta} = 4 \]

### Conclusion

The value of the expression is \(4\).