What is the value of `(sin 60^(@))/( cos^(2) 45^(@) ) - 3 tan 30^(@) + 5 cos 90^(@)`

Options

A. 1
B. `-1`
C. `(2)/(5)`
D. 0

Answer

To solve the expression \((\sin 60^\circ)/( \cos^2 45^\circ ) - 3 \tan 30^\circ + 5 \cos 90^\circ\), we will evaluate each trigonometric function step by step.

### Step 1: Calculate \(\sin 60^\circ\)
The value of \(\sin 60^\circ\) is:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]

### Step 2: Calculate \(\cos 45^\circ\)
The value of \(\cos 45^\circ\) is:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \]

### Step 3: Calculate \(\cos^2 45^\circ\)
Now we need to square \(\cos 45^\circ\):
\[ \cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 4: Substitute values into the expression
Now we can substitute the values into the expression:
\[ \frac{\sin 60^\circ}{\cos^2 45^\circ} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{2} \times \frac{2}{1} = \sqrt{3} \]

### Step 5: Calculate \(\tan 30^\circ\)
The value of \(\tan 30^\circ\) is:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]

### Step 6: Substitute \(\tan 30^\circ\) into the expression
Now, we substitute \(3 \tan 30^\circ\):
\[ 3 \tan 30^\circ = 3 \times \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \]

### Step 7: Calculate \(\cos 90^\circ\)
The value of \(\cos 90^\circ\) is:
\[ \cos 90^\circ = 0 \]
Thus, \(5 \cos 90^\circ = 5 \times 0 = 0\).

### Step 8: Combine all parts of the expression
Now we can combine all parts:
\[ \sqrt{3} - \sqrt{3} + 0 = 0 \]

### Final Answer
The final value of the expression is:
\[ \boxed{0} \]