If $\tan (\theta)=\frac{24}{7}, 0 \leq \theta \leq \frac{\pi}{2}$, then $\sin (\theta)$ equals $\cos (\theta)$ equals $\sec (\theta)$ equals

Step 1 ：We are given that \(\tan (\theta)=\frac{24}{7}\). We know that \(\tan (\theta)=\frac{\sin (\theta)}{\cos (\theta)}\). So, we can write \(\sin (\theta)=\tan (\theta) \cdot \cos (\theta)\).

Step 2 ：We also know that \(\sec (\theta)=\frac{1}{\cos (\theta)}\). So, we need to find the value of \(\cos (\theta)\) first.

Step 3 ：We can use the identity \(\tan^2 (\theta) + 1 = \sec^2 (\theta)\) to find the value of \(\cos (\theta)\).

Step 4 ：By substituting the given value of \(\tan (\theta)\) into the identity, we get \(\sec^2 (\theta) = (\frac{24}{7})^2 + 1 = 12.755102040816325\). Taking the square root of both sides, we get \(\sec (\theta) = 3.571428571428571\).

Step 5 ：Since \(\sec (\theta) = \frac{1}{\cos (\theta)}\), we can find \(\cos (\theta) = \frac{1}{\sec (\theta)} = 0.28\).

Step 6 ：Substituting the value of \(\cos (\theta)\) into the equation \(\sin (\theta)=\tan (\theta) \cdot \cos (\theta)\), we get \(\sin (\theta) = \frac{24}{7} \cdot 0.28 = 0.9600000000000001\).

Step 7 ：Now that we have the values of \(\cos (\theta)\), \(\sin (\theta)\), and \(\sec (\theta)\), we can check if they are equal.

Step 8 ：\(\cos (\theta) = 0.28\), \(\sin (\theta) = 0.9600000000000001\), and \(\sec (\theta) = 3.571428571428571\).

Step 9 ：\(\boxed{\text{Final Answer: The values of } \cos (\theta), \sin (\theta), \text{ and } \sec (\theta) \text{ are not equal. Therefore, the statement in the question is false.}}\)