Problem 35 - Algebra and Trigonometry

Question

Verify each identity. \(\frac{\sec x-\csc x}{\sec x+\csc x}=\frac{\tan x-1}{\tan x+1}\)

Step-by-Step Solution

Answer

The given identity \(\frac{\sec x-\csc x}{\sec x+csc x}=\frac{\tan x-1}{\tan x+1}\) is true as both sides simplify to \(\frac{\sin x - \cos x}{\sin x + \cos x}\)

Step 1: Rewrite the trigonometric functions in terms of sines and cosines

Rewrite sec x and csc x as \(1/\cos x\) and \(1/\sin x\) respectively and tan x as \(\sin x/\cos x\). Now the equation becomes \(\frac{(1/\cos x - 1/\sin x)}{(1/\cos x + 1/\sin x)}=\frac{(\sin x/\cos x - 1)}{(\sin x/\cos x + 1)}\).

Step 2: Simplify the equation

Cross-multiply the fractions in the left-hand side to get a common denominator which gives \(\frac{\sin x - \cos x}{\sin x + \cos x}\). Cross multiply the fractions in the Right-Hand side too, in order to get \(\frac{\sin x -\cos x}{\sin x +\cos x}\).

Step 3: Verify the identity

Now, it's confirmed that the left hand side is equal to the right hand side; \(\frac{\sin x - \cos x}{\sin x + \cos x}=\frac{\sin x -\cos x}{\sin x +\cos x}\) which verifies the identity.

Key Concepts

Sine and Cosine Functions

The sine and cosine functions are fundamental in trigonometry, representing the y-coordinate and x-coordinate of a point on the unit circle, respectively. These functions are periodic, meaning they repeat their values in regular intervals.

• The sine function, denoted as \(\sin x\), is the ratio of the opposite side to the hypotenuse in a right triangle.
• The cosine function, denoted as \(\cos x\), represents the ratio of the adjacent side to the hypotenuse.

An important aspect of these functions is their ability to be expressed in terms of each other using the Pythagorean identity:\[\sin^2 x + \cos^2 x = 1\]This identity is very helpful in simplifying trigonometric expressions and verifying identities. In problems involving trigonometric identities, rewriting functions in terms of sine and cosine is a crucial first step. It often leads to simpler forms and helps in verifying equations.
Understanding how sine and cosine relate to other trigonometric functions such as tangent, secant, and cosecant, can make it easier to handle complex trigonometric identities.

Secant and Cosecant Functions

The secant and cosecant functions are the reciprocals of the cosine and sine functions, respectively. Understanding these functions is key in solving many trigonometric problems.

• The secant function, \(\sec x\), is defined as \(\frac{1}{\cos x}\).
• The cosecant function, \(\csc x\), is defined as \(\frac{1}{\sin x}\).

These functions are less intuitive than sine and cosine, because they are related to ratios involving the entire hypotenuse of a right triangle as opposed to the adjacent or opposite sides. The reciprocal nature makes them particularly useful when working with trigonometric identities that involve division by sine or cosine.
In verifying trigonometric identities, converting secant and cosecant into their sine and cosine equivalents as was done in the exercise, simplifies the expressions into a more manageable form. This simplification is crucial for making comparisons and verifying whether identities hold true.

Trigonometric Simplification

Trigonometric simplification involves rewriting complex trigonometric expressions in simpler forms. This can help in solving equations, especially when verifying identities, as demonstrated in the original exercise.
Here are some key techniques for trigonometric simplification:

• Converting trigonometric functions to sine and cosine: Since all basic trigonometric functions can be expressed in terms of sine and cosine, doing this often reduces the complexity of equations.
• Finding a common denominator: When dealing with fractions, like in the identity provided, cross-multiplying to find a common denominator can simplify the expression significantly.
• Using basic identities: Employing identities such as the Pythagorean identity or reciprocal identities can be instrumental in reducing expressions.

It is often the combination of these techniques that leads to the successful verification of identities or the solution of trigonometric equations. In the exercise, simplifying both sides of the equation to \(\frac{\sin x - \cos x}{\sin x + \cos x}\) verified the identity. Mastering simplification can boost a student's confidence and efficiency in handling trigonometric problems.