If `cosecA=2`, then `(1)/(tanA)+(sinA)/(1+cosA)=?`

Options

A. 2
B. 3
C. 1
D. 4

Answer

To solve the problem, we start with the given information and simplify the expression step by step.

### Given:
\[ \csc A = 2 \]

### Step 1: Find \(\sin A\) and \(\cos A\)
Since \(\csc A = \frac{1}{\sin A}\), we can find \(\sin A\):
\[ \sin A = \frac{1}{\csc A} = \frac{1}{2} \]

Now, we can use the Pythagorean identity to find \(\cos A\):
\[ \sin^2 A + \cos^2 A = 1 \]
Substituting \(\sin A\):
\[ \left(\frac{1}{2}\right)^2 + \cos^2 A = 1 \]
\[ \frac{1}{4} + \cos^2 A = 1 \]
\[ \cos^2 A = 1 - \frac{1}{4} = \frac{3}{4} \]
\[ \cos A = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Step 2: Substitute into the Expression
Now we need to evaluate the expression:
\[ \frac{1}{\tan A} + \frac{\sin A}{1 + \cos A} \]
First, we find \(\tan A\):
\[ \tan A = \frac{\sin A}{\cos A} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \]
Thus,
\[ \frac{1}{\tan A} = \sqrt{3} \]

### Step 3: Evaluate \(\frac{\sin A}{1 + \cos A}\)
Now, substituting \(\sin A\) and \(\cos A\):
\[ 1 + \cos A = 1 + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2} \]
So,
\[ \frac{\sin A}{1 + \cos A} = \frac{\frac{1}{2}}{\frac{2 + \sqrt{3}}{2}} = \frac{1}{2 + \sqrt{3}} \]

### Step 4: Combine the Results
Now we combine the two parts:
\[ \sqrt{3} + \frac{1}{2 + \sqrt{3}} \]

### Step 5: Rationalize the Denominator
To combine these, we rationalize the denominator of \(\frac{1}{2 + \sqrt{3}}\):
\[ \frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3} \]

### Step 6: Final Combination
Now we can add:
\[ \sqrt{3} + (2 - \sqrt{3}) = 2 \]

### Final Answer:
\[ \frac{1}{\tan A} + \frac{\sin A}{1 + \cos A} = 2 \]

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