Solving Trigonometric Identities: sec A - tan A 1/3, Find sec A tan A

In trigonometry, we often encounter problems that require the application of fundamental identities. One such intriguing problem is: given that sec A - tan A 1/3, what is the value of sec A tan A?

Understanding the Problem

The problem provides us with a relationship between the secant and tangent of an angle A, specifically that sec A - tan A 1/3. We need to find the value of sec A tan A. This requires an understanding of trigonometric identities and algebraic manipulation.

Trigonometric Identities and Expressions

The primary trigonometric identities we will use are:

sec^2 A - tan^2 A 1 sec A 1/cos A and tan A sin A/cos A

These identities will serve as the foundation for our calculations.

Step-by-step Solution

Let's define:

x sec A tan A y sec A - tan A

We are given that:

y 1/3

Now, let's derive the relationship between x and y using the given expressions:

Multiplying the given equation by x:

x y sec A tan A (sec A - tan A)

Using the identity sec^2 A - tan^2 A 1:

x y sec^2 A - tan^2 A

Substitute the known identity:

x y 1

We also know that:

x - y sec A - tan A

Therefore:

x - (1/3) sec A - tan A

From the equations, we can express:

sec A (x y) / 2

tan A (x - y) / 2

With the given value of y, we substitute:

sec A - tan A 1/3

sec A - tan A (x - 1/3) / 2 1/3

solving for x, we get x 3

Thus, the value of sec A tan A is:

x 3

Conclusion

By using the trigonometric identities and algebraic manipulation, we have successfully found that sec A tan A 3, given that sec A - tan A 1/3. This problem illustrates the power of combining fundamental trigonometric identities with algebraic techniques.

Related Keywords and Trigonometric Identities

Trigonometric Identities: sec^2 A - tan^2 A 1 secA: sec A 1/cos A tanA: tan A sin A/cos A

Understanding these foundational concepts and their application in solving complex trigonometric problems will be invaluable for students and mathematicians alike.