Understanding the Trigonometry Problem

The problem asks us to find the value of $\tan A + \cot A$ given that $\cos A = \frac{63}{65}$. This is a standard trigonometry problem involving trigonometric ratios and identities.

Step-by-Step Solution to Find tan A + cot A

To find $\tan A$ and $\cot A$, we first need to determine the value of $\sin A$. We can use the fundamental trigonometric identity:

\(\sin^2 A + \cos^2 A = 1\)

We are given $\cos A = \frac{63}{65}$. Substitute this value into the identity:

\(\sin^2 A + \left(\frac{63}{65}\right)^2 = 1\)

\(\sin^2 A + \frac{69^2}{65^2} = 1\)

\(\sin^2 A + \frac{3969}{4225} = 1\)

Now, solve for \(\sin^2 A\):

\(\sin^2 A = 1 - \frac{3969}{4225}\)

To subtract, find a common denominator:

\(\sin^2 A = \frac{4225}{4225} - \frac{3969}{4225}\)

\(\sin^2 A = \frac{4225 - 3969}{4225}\)

\(\sin^2 A = \frac{256}{4225}\)

Now, take the square root to find $\sin A$. Assuming $A$ is in a quadrant where $\sin A$ is positive (like the first quadrant):

\(\sin A = \sqrt{\frac{256}{4225}}\)

\(\sin A = \frac{\sqrt{256}}{\sqrt{4225}}\)

\(\sin A = \frac{16}{65}\)

Now that we have $\sin A$ and $\cos A$, we can find $\tan A$ and $\cot A$.

Recall the definitions of $\tan A$ and $\cot A$ in terms of $\sin A$ and $\cos A$:

• \(\tan A = \frac{\sin A}{\cos A}\)
• \(\cot A = \frac{\cos A}{\sin A}\) or \(\cot A = \frac{1}{\tan A}\)

Using the values we found:

\(\tan A = \frac{\frac{16}{65}}{\frac{63}{65}} = \frac{16}{65} \times \frac{65}{63} = \frac{16}{63}\)

And

\(\cot A = \frac{1}{\tan A} = \frac{1}{\frac{16}{63}} = \frac{63}{16}\)

Finally, we need to find the value of $\tan A + \cot A$:

\(\tan A + \cot A = \frac{16}{63} + \frac{63}{16}\)

To add these fractions, find a common denominator, which is $63 \times 16 = 1008$.

\(\frac{16}{63} + \frac{63}{16} = \frac{16 \times 16}{63 \times 16} + \frac{63 \times 63}{16 \times 63}\)

\(\frac{16}{63} + \frac{63}{16} = \frac{256}{1008} + \frac{3969}{1008}\)

\(\tan A + \cot A = \frac{256 + 3969}{1008}\)

\(\tan A + \cot A = \frac{4225}{1008}\)

Now, convert the fraction to a decimal and round to two decimal places:

\(\frac{4225}{1008} \approx 4.19146...\)

Rounding to two decimal places, we get approximately 4.19.

Calculation Summary

Conclusion on tan A + cot A Value

Based on our calculations using the given value of \(\cos A\), the value of \(\tan A + \cot A\) is approximately 4.19 when rounded to two decimal places.

Revision Table: Trigonometric Ratios and Identities

Additional Information: Pythagorean Identity

The identity \( \sin^2 A + \cos^2 A = 1 \) is a core Pythagorean identity in trigonometry. It comes directly from the Pythagorean theorem applied to a right-angled triangle with hypotenuse 1. If the angle is $A$, the adjacent side is \(\cos A\) and the opposite side is \(\sin A\). The theorem states \( (\text{opposite})^2 + (\text{adjacent})^2 = (\text{hypotenuse})^2 \), which translates to \( (\sin A)^2 + (\cos A)^2 = 1^2 \), or \( \sin^2 A + \cos^2 A = 1 \). This identity is fundamental for finding missing trigonometric ratios when one is known, as demonstrated in this problem involving \( \cos A \), \( \sin A \), \( \tan A \), and \( \cot A \).