Question

Let \(\mathrm{f}(\mathrm{x})=\sec \mathrm{x}+\tan \mathrm{x}, \mathrm{g}(\mathrm{x})=[(\tan \mathrm{x}) /\\{1-\sec (\mathrm{x})\\}]\) Statement \(-1 \mathrm{~g}\) is an odd function Statement \(-2 \mathrm{f}\) is neither an odd function nor an even function (a) Statement lis true (b) Statement 2 is true (c) 1 and 2 both are true (d) 1 and 2 both are false

Short answer

(a) Statement 1 is true

Step-by-step solution

Check oddness for g(x)

We will find \(g(-x)\) and see if it is equal to \(-g(x)\) : \(g(x) = \frac{\tan x}{1 - \sec x}\) \(g(-x) = \frac{\tan(-x)}{1 - \sec(-x)} = \frac{-\tan x}{1 - \sec x}\) Now we'll compare this with \(-g(x)\): \(-g(x) = -\frac{\tan x}{1 - \sec x}\) Since \(g(-x) = -g(x)\), g(x) is an odd function. Thus, Statement 1 is true.

Check oddness and evenness for f(x)

We will find \(f(-x)\) and compare it with both \(f(x)\) and \(-f(x)\): \(f(x) = \sec x + \tan x\) \(f(-x) = \sec(-x) + \tan(-x) = -\sec x - \tan x = -\left(\sec x + \tan x \right) = -f(x)\) Since \(f(-x) = -f(x)\), f(x) is an odd function. Thus, Statement 2 is false. f(x) is an odd function. #Conclusion# As Statement 1 is true and Statement 2 is false, the correct option is: (a) Statement 1 is true

Key concepts

Trigonometric Identities

Mastering trigonometric identities is essential for solving many problems in mathematics, especially when dealing with trigonometric functions. A trigonometric identity is a mathematical statement that equates one trigonometric expression to another, and these identities hold for all values of the variable where both sides of the equation are defined.

Common examples include the Pythagorean identities, such as \( \sin^2 x + \cos^2 x = 1 \), and angle sum and difference identities, such as \( \sin(x ± y) = \sin x \cos y ± \cos x \sin y \). Identities are used to simplify expressions, transform the form of an equation, and even to prove other mathematical properties. For instance, understanding that \( \tan x = \frac{\sin x}{\cos x} \) and that \( \sec x = \frac{1}{\cos x} \) helps us manipulate trigonometric functions and solve complex equations with more ease.

Function Symmetry

Function symmetry refers to the property where a function exhibits a mirror-like behavior across a line or point. The most common types of symmetries are odd and even symmetry, associated with the origin and the y-axis, respectively.

An even function, satisfying \( f(x) = f(-x) \), is symmetric about the y-axis. It looks the same on both the left and right sides of the y-axis. A classic example is the cosine function. In contrast, an odd function, which satisfies \( f(-x) = -f(x) \), is symmetric about the origin, meaning if you rotate its graph 180 degrees around the origin, you'll get the same graph. The sine function is a typical odd function.

Not every function is purely odd or even; some may exhibit no symmetry or have a more complex type of symmetry not classified under these basic types.

Trigonometric Functions

The trigonometric functions are fundamental in mathematics, defining relationships between the angles and sides of triangles, and they have applications in many fields such as physics, engineering, and surveying. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), and their reciprocals, cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)).

Each function has a specific definition based on the context of a right-angled triangle or a unit circle. For example, in a right triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse, while on the unit circle, it's the y-coordinate of the corresponding point on the circle. Trigonometric functions also have unique graphs that exhibit periodic behavior and their properties have profound implications in studying waves and oscillations.