If `cosec theta=(29)/(21)`,where `0 lt theta lt 90^@`, then what is the value of `4sec theta + 4tan theta` ?

Options

A. 5
B. 10
C. 15
D. 20

Answer

To solve the problem, we need to find the value of \(4 \sec \theta + 4 \tan \theta\) given that \(\csc \theta = \frac{29}{21}\).

### Step 1: Understand the relationship between cosecant, secant, and tangent.
We know that:
- \(\csc \theta = \frac{1}{\sin \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

### Step 2: Find the values of sine and cosine.
Given \(\csc \theta = \frac{29}{21}\), we can find \(\sin \theta\):
\[ \sin \theta = \frac{1}{\csc \theta} = \frac{21}{29} \]

### Step 3: Use the Pythagorean identity to find \(\cos \theta\).
Using the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \(\sin \theta\):
\[ \left(\frac{21}{29}\right)^2 + \cos^2 \theta = 1 \]
Calculating \(\left(\frac{21}{29}\right)^2\):
\[ \frac{441}{841} + \cos^2 \theta = 1 \]
\[ \cos^2 \theta = 1 - \frac{441}{841} = \frac{841 - 441}{841} = \frac{400}{841} \]
Taking the square root:
\[ \cos \theta = \frac{20}{29} \]

### Step 4: Calculate \(\sec \theta\) and \(\tan \theta\).
Now we can find \(\sec \theta\) and \(\tan \theta\):
\[ \sec \theta = \frac{1}{\cos \theta} = \frac{29}{20} \]
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{21}{29}}{\frac{20}{29}} = \frac{21}{20} \]

### Step 5: Substitute into the expression \(4 \sec \theta + 4 \tan \theta\).
Now we can substitute these values into the expression:
\[ 4 \sec \theta + 4 \tan \theta = 4 \left(\frac{29}{20}\right) + 4 \left(\frac{21}{20}\right) \]
Factoring out the common term:
\[ = 4 \left(\frac{29 + 21}{20}\right) = 4 \left(\frac{50}{20}\right) = 4 \times 2.5 = 10 \]

### Final Answer:
Thus, the value of \(4 \sec \theta + 4 \tan \theta\) is \(10\).

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