The value of `(sin 30^(@) - cos 60^(@) + tan 45^(@))/(cos 90^(@) + tan 45^(@) + sin 90^(@))` is

Options

A. `(1)/(2)`
B. 1
C. `sqrt(3)`
D. `infty`

Answer

To solve the expression \((\sin 30^\circ - \cos 60^\circ + \tan 45^\circ)/(\cos 90^\circ + \tan 45^\circ + \sin 90^\circ)\), we will evaluate the trigonometric ratios step by step.

### Step 1: Evaluate \(\sin 30^\circ\)
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]

### Step 2: Evaluate \(\cos 60^\circ\)
We know that:
\[ \cos 60^\circ = \frac{1}{2} \]

### Step 3: Evaluate \(\tan 45^\circ\)
We know that:
\[ \tan 45^\circ = 1 \]

### Step 4: Evaluate \(\cos 90^\circ\)
We know that:
\[ \cos 90^\circ = 0 \]

### Step 5: Evaluate \(\sin 90^\circ\)
We know that:
\[ \sin 90^\circ = 1 \]

### Step 6: Substitute the values into the expression
Now we substitute the evaluated values into the expression:
\[ \frac{\sin 30^\circ - \cos 60^\circ + \tan 45^\circ}{\cos 90^\circ + \tan 45^\circ + \sin 90^\circ} = \frac{\frac{1}{2} - \frac{1}{2} + 1}{0 + 1 + 1} \]

### Step 7: Simplify the numerator
The numerator simplifies as follows:
\[ \frac{1}{2} - \frac{1}{2} + 1 = 0 + 1 = 1 \]

### Step 8: Simplify the denominator
The denominator simplifies as follows:
\[ 0 + 1 + 1 = 2 \]

### Step 9: Final calculation
Now we can simplify the entire expression:
\[ \frac{1}{2} = \frac{1}{2} \]

Thus, the value of the expression is:
\[ \frac{1}{2} \]

### Final Answer:
The value of the expression is \(\frac{1}{2}\).

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