To find the value of \(\cot\theta\) given \(\cos\theta = \frac{24}{25}\), we will use trigonometric identities and relationships. The formulae we will use are:

• \(\sin^2\theta + \cos^2\theta = 1\)
• \(\cot\theta = \frac{\cos\theta}{\sin\theta}\)

Step 1: Calculate \(\sin\theta\) using the Pythagorean identity:

Since we know that:

\(\cos\theta = \frac{24}{25}\)

We can substitute into \(\sin^2\theta + \cos^2\theta = 1\):

\(\sin^2\theta + \left(\frac{24}{25}\right)^2 = 1\)

\(\sin^2\theta + \frac{576}{625} = 1\)

\(\sin^2\theta = 1 - \frac{576}{625}\)

\(\sin^2\theta = \frac{625 - 576}{625}\)

\(\sin^2\theta = \frac{49}{625}\)

Taking the square root, since we assume \(\theta\) is in the first quadrant where \(\sin\theta\) is positive:

\(\sin\theta = \frac{7}{25}\)

Step 2: Calculate \(\cot\theta\) using the relation \(\cot\theta = \frac{\cos\theta}{\sin\theta}\):

\(\cot\theta = \frac{\frac{24}{25}}{\frac{7}{25}}\)

\(\cot\theta = \frac{24}{25} \times \frac{25}{7}\)

\(\cot\theta = \frac{24}{7}\)

Therefore, the value of \(\cot\theta\) is \(\frac{24}{7}\).