`2 cosec^2 23^@ cot^2 67^@ - sin^2 23^@-sin^2 67^@-cot^2 67^@` is equal to

Options

A. `sec^2 23^@`
B. `tan^2 23^@`
C. 0
D. 1

Answer

To solve the expression \( 2 \csc^2 23^\circ \cot^2 67^\circ - \sin^2 23^\circ - \sin^2 67^\circ - \cot^2 67^\circ \), we will follow these steps:

### Step 1: Rewrite the trigonometric functions
We know that:
- \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \)
- \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

Using these identities, we can rewrite \( \csc^2 23^\circ \) and \( \cot^2 67^\circ \):
\[ \csc^2 23^\circ = \frac{1}{\sin^2 23^\circ} \]
\[ \cot^2 67^\circ = \cot^2 (90^\circ - 23^\circ) = \tan^2 23^\circ = \frac{\sin^2 23^\circ}{\cos^2 23^\circ} \]

### Step 2: Substitute into the expression
Substituting these into the expression gives:
\[ 2 \left(\frac{1}{\sin^2 23^\circ}\right) \left(\frac{\sin^2 23^\circ}{\cos^2 23^\circ}\right) - \sin^2 23^\circ - \sin^2 67^\circ - \cot^2 67^\circ \]
This simplifies to:
\[ 2 \left(\frac{1}{\cos^2 23^\circ}\right) - \sin^2 23^\circ - \sin^2 67^\circ - \tan^2 23^\circ \]

### Step 3: Simplify \( \sin^2 67^\circ \)
Using the identity \( \sin(90^\circ - \theta) = \cos \theta \):
\[ \sin^2 67^\circ = \cos^2 23^\circ \]

### Step 4: Substitute \( \sin^2 67^\circ \) back into the expression
Now we substitute \( \sin^2 67^\circ \):
\[ 2 \sec^2 23^\circ - \sin^2 23^\circ - \cos^2 23^\circ - \tan^2 23^\circ \]

### Step 5: Use the Pythagorean identity
We know that:
\[ \sin^2 23^\circ + \cos^2 23^\circ = 1 \]
Thus, we can replace \( \sin^2 23^\circ + \cos^2 23^\circ \) with \( 1 \):
\[ 2 \sec^2 23^\circ - 1 - \tan^2 23^\circ \]

### Step 6: Use the identity \( \sec^2 \theta = 1 + \tan^2 \theta \)
Using the identity:
\[ \sec^2 23^\circ = 1 + \tan^2 23^\circ \]
Substituting this into the expression gives:
\[ 2(1 + \tan^2 23^\circ) - 1 - \tan^2 23^\circ \]
This simplifies to:
\[ 2 + 2\tan^2 23^\circ - 1 - \tan^2 23^\circ = 1 + \tan^2 23^\circ \]

### Step 7: Final result
Thus, the final result is:
\[ \boxed{1 + \tan^2 23^\circ} \]