The value of `(sec^(2)45^@-cot^(2)45^@)-(sin^(2)30^@+sin^(2) 60^(@))` is

Options

A. 1
B. `2sqrt3`
C. 0
D. `1/sqrt2`

Answer

To solve the expression `(sec^(2)45° - cot^(2)45°) - (sin^(2)30° + sin^(2)60°)`, we will evaluate each component step by step.

### Step 1: Calculate `sec^(2)45°`
The secant function is the reciprocal of the cosine function.
- We know that `cos(45°) = 1/√2`.
- Therefore, `sec(45°) = 1/cos(45°) = √2`.
- Now, squaring this gives us:
\[ sec^{2}(45°) = (√2)^{2} = 2. \]

### Step 2: Calculate `cot^(2)45°`
The cotangent function is the reciprocal of the tangent function.
- We know that `tan(45°) = 1`.
- Therefore, `cot(45°) = 1/tan(45°) = 1`.
- Now, squaring this gives us:
\[ cot^{2}(45°) = (1)^{2} = 1. \]

### Step 3: Calculate `sec^(2)45° - cot^(2)45°`
Now we can substitute the values we calculated:
\[ sec^{2}(45°) - cot^{2}(45°) = 2 - 1 = 1. \]

### Step 4: Calculate `sin^(2)30°`
We know that:
- `sin(30°) = 1/2`.
- Therefore, squaring this gives us:
\[ sin^{2}(30°) = (1/2)^{2} = 1/4. \]

### Step 5: Calculate `sin^(2)60°`
We know that:
- `sin(60°) = √3/2`.
- Therefore, squaring this gives us:
\[ sin^{2}(60°) = (√3/2)^{2} = 3/4. \]

### Step 6: Calculate `sin^(2)30° + sin^(2)60°`
Now we can add the two results:
\[ sin^{2}(30°) + sin^{2}(60°) = 1/4 + 3/4 = 4/4 = 1. \]

### Step 7: Combine the results
Now we substitute back into the original expression:
\[ (sec^{2}(45°) - cot^{2}(45°)) - (sin^{2}(30°) + sin^{2}(60°)) = 1 - 1 = 0. \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{0}. \]