`( Sec^2 theta - cot ^2 (90 - theta))/( cosec ^2 67^@ - tan ^2 23^@) + sin ^2 40 ^@ + sin ^2 50^@ ` is equal to

Options

A. 0
B. 4
C. 2
D. 1

Answer

To solve the expression

\[ \frac{\sec^2 \theta - \cot^2 (90^\circ - \theta)}{\csc^2 67^\circ - \tan^2 23^\circ} + \sin^2 40^\circ + \sin^2 50^\circ, \]

we will simplify it step by step.

### Step 1: Simplify \(\cot^2(90^\circ - \theta)\)

Using the co-function identity, we know that:

\[ \cot(90^\circ - \theta) = \tan \theta. \]

Thus,

\[ \cot^2(90^\circ - \theta) = \tan^2 \theta. \]

### Step 2: Rewrite the numerator

Now, we can rewrite the numerator:

\[ \sec^2 \theta - \cot^2(90^\circ - \theta) = \sec^2 \theta - \tan^2 \theta. \]

### Step 3: Use the identity \(\sec^2 \theta - \tan^2 \theta = 1\)

From trigonometric identities, we have:

\[ \sec^2 \theta - \tan^2 \theta = 1. \]

So, the numerator simplifies to:

\[ 1. \]

### Step 4: Simplify the denominator \(\csc^2 67^\circ - \tan^2 23^\circ\)

Using the identity \(\csc(90^\circ - x) = \sec x\):

\[ \csc 67^\circ = \sec(90^\circ - 67^\circ) = \sec 23^\circ. \]

Thus,

\[ \csc^2 67^\circ = \sec^2 23^\circ. \]

Now, we can rewrite the denominator:

\[ \csc^2 67^\circ - \tan^2 23^\circ = \sec^2 23^\circ - \tan^2 23^\circ. \]

### Step 5: Use the identity \(\sec^2 x - \tan^2 x = 1\)

Again, using the identity:

\[ \sec^2 23^\circ - \tan^2 23^\circ = 1. \]

So, the denominator simplifies to:

\[ 1. \]

### Step 6: Substitute back into the expression

Now we can substitute back into the original expression:

\[ \frac{1}{1} + \sin^2 40^\circ + \sin^2 50^\circ. \]

This simplifies to:

\[ 1 + \sin^2 40^\circ + \sin^2 50^\circ. \]

### Step 7: Use the identity \(\sin^2 x + \cos^2 x = 1\)

Notice that:

\[ \sin^2 50^\circ = \cos^2 40^\circ. \]

Thus,

\[ \sin^2 40^\circ + \sin^2 50^\circ = \sin^2 40^\circ + \cos^2 40^\circ = 1. \]

### Step 8: Final simplification

Now we can add:

\[ 1 + 1 = 2. \]

### Conclusion

Thus, the entire expression simplifies to:

\[ \boxed{2}. \]