Find the value of `sin^2 60^@+tan^2 45^@+cosec30^@xxsec60^@`:

Options

A. 1
B. `16/5`
C. -1
D. `23/4`

Answer

To solve the expression \( \sin^2 60^\circ + \tan^2 45^\circ + \csc 30^\circ \times \sec 60^\circ \), we will evaluate each trigonometric function step by step.

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

We know that:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]
Thus,
\[ \sin^2 60^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 2: Calculate \( \tan^2 45^\circ \)

We know that:
\[ \tan 45^\circ = 1 \]
Thus,
\[ \tan^2 45^\circ = 1^2 = 1 \]

### Step 3: Calculate \( \csc 30^\circ \)

We know that:
\[ \csc 30^\circ = \frac{1}{\sin 30^\circ} = \frac{1}{\frac{1}{2}} = 2 \]

### Step 4: Calculate \( \sec 60^\circ \)

We know that:
\[ \sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{\frac{1}{2}} = 2 \]

### Step 5: Combine \( \csc 30^\circ \) and \( \sec 60^\circ \)

Now we can calculate:
\[ \csc 30^\circ \times \sec 60^\circ = 2 \times 2 = 4 \]

### Step 6: Add all the values together

Now we can combine all the results:
\[ \sin^2 60^\circ + \tan^2 45^\circ + \csc 30^\circ \times \sec 60^\circ = \frac{3}{4} + 1 + 4 \]

To add these, we convert \( 1 \) and \( 4 \) into fractions with a common denominator of 4:
\[ 1 = \frac{4}{4}, \quad 4 = \frac{16}{4} \]
Thus,
\[ \frac{3}{4} + \frac{4}{4} + \frac{16}{4} = \frac{3 + 4 + 16}{4} = \frac{23}{4} \]

### Final Answer
The value of the expression is:
\[ \frac{23}{4} \]

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