The value of `(sin^2""7""1/2""+cos^2""7""1/2""^@)-(sin^2""30^@+cos^2""30^@)+(sin^2""7^@+sin^2""83^@)` is equal to

Options

A. 3
B. `3""1/2`
C. 2
D. 1

Answer

To solve the expression \((\sin^2 71.5^\circ + \cos^2 71.5^\circ) - (\sin^2 30^\circ + \cos^2 30^\circ) + (\sin^2 7^\circ + \sin^2 83^\circ)\), we can break it down step by step.

### Step 1: Use the Pythagorean Identity
We know that for any angle \( \theta \):
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
So, we can apply this identity to the first two terms.

**Calculation:**
\[ \sin^2 71.5^\circ + \cos^2 71.5^\circ = 1 \]
\[ \sin^2 30^\circ + \cos^2 30^\circ = 1 \]

### Step 2: Substitute the Values
Now, substitute these values back into the expression:
\[ (1) - (1) + (\sin^2 7^\circ + \sin^2 83^\circ) \]

### Step 3: Simplify the Expression
This simplifies to:
\[ 0 + (\sin^2 7^\circ + \sin^2 83^\circ) \]

### Step 4: Use the Identity for Sine
We know that:
\[ \sin(90^\circ - \theta) = \cos \theta \]
Thus, we can express \(\sin^2 83^\circ\) in terms of \(\sin^2 7^\circ\):
\[ \sin^2 83^\circ = \sin^2 (90^\circ - 7^\circ) = \cos^2 7^\circ \]

### Step 5: Combine the Terms
Now we can rewrite the expression:
\[ \sin^2 7^\circ + \sin^2 83^\circ = \sin^2 7^\circ + \cos^2 7^\circ \]

### Step 6: Apply the Pythagorean Identity Again
Using the Pythagorean identity again:
\[ \sin^2 7^\circ + \cos^2 7^\circ = 1 \]

### Final Result
Thus, the final value of the entire expression is:
\[ 1 \]

### Summary
The value of \((\sin^2 71.5^\circ + \cos^2 71.5^\circ) - (\sin^2 30^\circ + \cos^2 30^\circ) + (\sin^2 7^\circ + \sin^2 83^\circ)\) is equal to \(1\).

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