Evaluating Trigonometric Expression with Given Tan Value

We are given that $\tan\theta = \frac{5}{12}$ and the angle $\theta$ lies in the first quadrant ($0 < \theta < \frac{\pi}{2}$). We need to find the value of the expression $\frac{\cos\theta + 5\cot\theta}{\text{cosec}\theta - \cos\theta}$.

Calculating Required Trigonometric Values

Since $\theta$ is in the first quadrant, all trigonometric ratios will be positive. We can imagine a right-angled triangle where the opposite side is 5 and the adjacent side is 12.

• Calculate the hypotenuse ($h$): $h = \sqrt{\text{opposite}^2 + \text{adjacent}^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.
• Find $\cos\theta$: $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}$.
• Find $\cot\theta$: $\cot\theta = \frac{1}{\tan\theta} = \frac{1}{\frac{5}{12}} = \frac{12}{5}$.
• Find $\sin\theta$ first: $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13}$.
• Find $\text{cosec}\theta$: $\text{cosec}\theta = \frac{1}{\sin\theta} = \frac{1}{\frac{5}{13}} = \frac{13}{5}$.

Substituting and Simplifying the Expression

Now substitute these values into the given expression:

$ \frac{\cos\theta + 5\cot\theta}{\text{cosec}\theta - \cos\theta} = \frac{\frac{12}{13} + 5 \times \frac{12}{5}}{\frac{13}{5} - \frac{12}{13}} $

Simplify the numerator and the denominator separately:

• Numerator: $ \frac{12}{13} + 5 \times \frac{12}{5} = \frac{12}{13} + 12 = \frac{12 + 12 \times 13}{13} = \frac{12 + 156}{13} = \frac{168}{13} $
• Denominator: $ \frac{13}{5} - \frac{12}{13} = \frac{13 \times 13 - 12 \times 5}{5 \times 13} = \frac{169 - 60}{65} = \frac{109}{65} $

Now, divide the simplified numerator by the simplified denominator:

$ \frac{\frac{168}{13}}{\frac{109}{65}} = \frac{168}{13} \times \frac{65}{109} = \frac{168 \times 5}{109} = \frac{840}{109} $

Final Result

The value of the expression $\frac{\cos\theta + 5\cot\theta}{\text{cosec}\theta - \cos\theta}$ is $\frac{840}{109}$.