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Deriving sec x tan x Given tan(x/2) r
Deriving sec x tan x Given tan(x/2) r
In this article, we will explore a specific trigonometric problem and demonstrate a step-by-step solution to it. Given tan(x/2) r, we aim to derive the expression for sec x tan x. We will use various trigonometric identities and formulas to achieve this goal. This explanation is valuable for students and professionals interested in understanding and solving trigonometric equations.
Trigonometric Identities and Formulas
First, let's list the essential trigonometric identities and formulas we will use:
Double Angle Formulas: These formulas express trigonometric functions of double angles in terms of single angles. Basic Trigonometric Functions: Definitions and relationships between sine, cosine, and tangent functions. Pythagorean Identity: sin^2 x cos^2 x 1, which helps in expressing trigonometric functions in different forms.Step-by-Step Solution
Step 1: Express tan x in terms of r
Given that tan(x/2) r, we can express tan x as follows:
tan x frac{2 tan frac{x}{2}}{1 - tan^2 frac{x}{2}} frac{2r}{1 - r^2}
Step 2: Compute sec x
We need to find a direct expression for sec x. Using the Pythagorean identity and the definition of sec x:
sec x frac{1}{cos x} sqrt{1 tan^2 x}
Substituting the expression for tan x into the Pythagorean identity:
tan^2 x left(frac{2r}{1 - r^2}right)^2 frac{4r^2}{(1 - r^2)^2}
Hence,
sec x sqrt{1 frac{4r^2}{(1 - r^2)^2}} frac{sqrt{1 - r^2^2 4r^2}}{1 - r^2}
Step 3: Compute sec x tan x
Now that we have expressions for both sec x and tan x, we can find sec x tan x:
sec x tan x frac{1}{sec x} tan x frac{sqrt{1 - r^2^2 4r^2} times 2r}{1 - r^2}
This results in the final expression:
sec x tan x frac{sqrt{1 - r^2^2 4r^2} times 2r}{1 - r^2}
Conclusion
We have derived an important expression using given trigonometric identities and formulas, showing how to compute sec x tan x given tan(x/2) r. Understanding these processes is crucial for solving more complex trigonometric problems and formulating clearer solutions.