Let's calculate the given expression step-by-step.
### Given Expression:
[tex]\[ \frac{\tan 45^{\circ}}{\operatorname{cosec}30^{\circ}} + \frac{\sec 60^{\circ}}{\cot 45^{\circ}} + \frac{2 \sin 9^{\circ}}{\cos 0^{\circ}} \][/tex]
### Step 1: Calculate [tex]\(\tan 45^{\circ}\)[/tex]
The tangent of [tex]\(45^\circ\)[/tex] is:
[tex]\[ \tan 45^\circ = 1 \][/tex]
### Step 2: Calculate [tex]\(\operatorname{cosec}30^{\circ}\)[/tex]
Cosecant is the reciprocal of sine. The sine of [tex]\(30^\circ\)[/tex] is [tex]\( \frac{1}{2} \)[/tex], so:
[tex]\[ \operatorname{cosec}30^\circ = \frac{1}{\sin 30^\circ} = \frac{1}{\frac{1}{2}} = 2 \][/tex]
### Step 3: Calculate [tex]\(\sec 60^{\circ}\)[/tex]
Secant is the reciprocal of cosine. The cosine of [tex]\(60^\circ\)[/tex] is [tex]\( \frac{1}{2} \)[/tex], so:
[tex]\[ \sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{\frac{1}{2}} = 2 \][/tex]
### Step 4: Calculate [tex]\(\cot 45^{\circ}\)[/tex]
Cotangent is the reciprocal of tangent. The tangent of [tex]\(45^\circ\)[/tex] is [tex]\(1\)[/tex], so:
[tex]\[ \cot 45^\circ = \frac{1}{\tan 45^\circ} = \frac{1}{1} = 1 \][/tex]
### Step 5: Calculate [tex]\(\sin 9^{\circ}\)[/tex]
The sine of [tex]\(9^\circ\)[/tex] is a small decimal:
[tex]\[ \sin 9^\circ \approx 0.15643446504023087 \][/tex]
### Step 6: Calculate [tex]\(\cos 0^{\circ}\)[/tex]
The cosine of [tex]\(0^\circ\)[/tex] is:
[tex]\[ \cos 0^\circ = 1 \][/tex]
### Step 7: Calculate each term in the expression
1. [tex]\(\frac{\tan 45^\circ}{\operatorname{cosec}30^\circ}\)[/tex]:
[tex]\[ \frac{\tan 45^\circ}{\operatorname{cosec}30^\circ} = \frac{1}{2} = 0.5 \][/tex]
2. [tex]\(\frac{\sec 60^\circ}{\cot 45^\circ}\)[/tex]:
[tex]\[ \frac{\sec 60^\circ}{\cot 45^\circ} = \frac{2}{1} = 2 \][/tex]
3. [tex]\(\frac{2 \sin 9^\circ}{\cos 0^\circ}\)[/tex]:
[tex]\[ \frac{2 \sin 9^\circ}{\cos 0^\circ} = \frac{2 \times 0.15643446504023087}{1} \approx 0.31286893008046174 \][/tex]
### Step 8: Add the results
Summing up the three terms:
[tex]\[ 0.5 + 2 + 0.31286893008046174 \approx 2.812868930080461 \][/tex]
Therefore, the value of the expression is:
[tex]\[ \boxed{2.812868930080461} \][/tex]
