If $\tan \theta =\frac{7}{24}$, then find the value of $({\cos }^2\theta -{\sin }^2\theta )$.

The correct answer is 527/625.

Step 1: Understand the given trigonometric ratio
We are given the value of tangent function as:

$\tan \theta =\frac{7}{24}$

Recall that in a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side:

$\tan \theta =\frac{\text{Opposite}}{\text{Adjacent}}$

Therefore, we can represent the lengths of the sides as:
Opposite side = 7
Adjacent side = 24

Step 2: Find the hypotenuse using the Pythagorean theorem
By the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:

${\text{Hypotenuse}}^2={\text{Opposite}}^2+{\text{Adjacent}}^2$

Substituting the values:

$\text{Hypotenuse}=\sqrt{7^2+{24}^2}$

$\text{Hypotenuse}=\sqrt{49+576}$

$\text{Hypotenuse}=\sqrt{625}=25$

Step 3: Determine the values of sine and cosine
Now we can write the values for sine and cosine using their respective definitions:

$\sin \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}=\frac{7}{25}$

$\cos \theta =\frac{\text{Adjacent}}{\text{Hypotenuse}}=\frac{24}{25}$

Step 4: Calculate the required expression
Substitute the values of sine and cosine into the expression

${\cos }^2\theta -{\sin }^2\theta$

This gives:

${\cos }^2\theta -{\sin }^2\theta ={\left(\frac{24}{25}\right)}^2-{\left(\frac{7}{25}\right)}^2$

${\cos }^2\theta -{\sin }^2\theta =\frac{576}{625}-\frac{49}{625}$

${\cos }^2\theta -{\sin }^2\theta =\frac{576-49}{625}=\frac{527}{625}$