The value of
`(tan ^(2) 60^(@) + 4 sin ^(2) 45 ^(@) + 3 sec ^(2) 30^(@) + 5 cos ^(2) 90^(@))/( "cosec" 30^(@) + sec 60^(@) - cot ^(2) 30^(@))` is

Options

A. 5
B. 3
C. 9
D. 2

Answer

To solve the expression

\[ \frac{\tan^2 60^\circ + 4 \sin^2 45^\circ + 3 \sec^2 30^\circ + 5 \cos^2 90^\circ}{\csc 30^\circ + \sec 60^\circ - \cot^2 30^\circ} \]

we will evaluate each trigonometric function step by step.

### Step 1: Calculate \(\tan^2 60^\circ\)

\[ \tan 60^\circ = \sqrt{3} \implies \tan^2 60^\circ = (\sqrt{3})^2 = 3 \]

### Step 2: Calculate \(4 \sin^2 45^\circ\)

\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \implies \sin^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
\[ 4 \sin^2 45^\circ = 4 \times \frac{1}{2} = 2 \]

### Step 3: Calculate \(3 \sec^2 30^\circ\)

\[ \sec 30^\circ = \frac{2}{\sqrt{3}} \implies \sec^2 30^\circ = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3} \]
\[ 3 \sec^2 30^\circ = 3 \times \frac{4}{3} = 4 \]

### Step 4: Calculate \(5 \cos^2 90^\circ\)

\[ \cos 90^\circ = 0 \implies \cos^2 90^\circ = 0^2 = 0 \]
\[ 5 \cos^2 90^\circ = 5 \times 0 = 0 \]

### Step 5: Combine the results in the numerator

\[ \tan^2 60^\circ + 4 \sin^2 45^\circ + 3 \sec^2 30^\circ + 5 \cos^2 90^\circ = 3 + 2 + 4 + 0 = 9 \]

### Step 6: Calculate the denominator

#### Step 6.1: Calculate \(\csc 30^\circ\)

\[ \csc 30^\circ = \frac{1}{\sin 30^\circ} = \frac{1}{\frac{1}{2}} = 2 \]

#### Step 6.2: Calculate \(\sec 60^\circ\)

\[ \sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{\frac{1}{2}} = 2 \]

#### Step 6.3: Calculate \(\cot^2 30^\circ\)

\[ \cot 30^\circ = \frac{1}{\tan 30^\circ} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \implies \cot^2 30^\circ = (\sqrt{3})^2 = 3 \]

### Step 7: Combine the results in the denominator

\[ \csc 30^\circ + \sec 60^\circ - \cot^2 30^\circ = 2 + 2 - 3 = 1 \]

### Step 8: Final calculation

Now, substituting back into the expression:

\[ \frac{9}{1} = 9 \]

Thus, the value of the expression is

\[ \boxed{9} \]