Problem 26 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\tan x}{\sec x}=\sin x $$

Step-by-Step Solution

Answer

The identity \( \frac{\tan x}{\sec x} = \sin x \) is verified by simplifying the left side to \( \sin x \).

Step 1: Understanding the Identity to Verify

The identity given is \( \frac{\tan x}{\sec x} = \sin x \). Our goal is to show that the left-hand side (LHS) simplifies to the right-hand side (RHS) \( \sin x \).

Step 1: Express \( \tan x \) and \( \sec x \) in terms of \( \sin x \) and \( \cos x \)

Recall that \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \). Substitute these into the LHS: \[ \frac{\tan x}{\sec x} = \frac{\frac{\sin x}{\cos x}}{\frac{1}{\cos x}} \].

Step 2: Simplify the Expression

Simplify the resulting complex fraction: \[ \frac{\frac{\sin x}{\cos x}}{\frac{1}{\cos x}} = \left(\frac{\sin x}{\cos x}\right) \times \left(\cos x\right) = \sin x \].

Step 3: Conclusion

The expression simplifies to \( \sin x \), which matches the right-hand side of the identity. Thus, the identity is verified.

Key Concepts

Trigonometric functions

Trigonometric functions are fundamental to understanding angles and their relationships within triangles and on the unit circle. In this context, they help us manipulate and verify identities. The key functions include the sine (\(\sin x\)), cosine (\(\cos x\)), tangent (\(\tan x\)), and secant (\(\sec x\)) functions. Each of these has specific mathematical definitions:

• Sine (\(\sin x\)) is the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (\(\cos x\)) is the ratio of the adjacent side to the hypotenuse.
• Tangent (\(\tan x\)) is the ratio of the sine to the cosine of an angle, which can be expressed as \(\frac{\sin x}{\cos x}\).
• Secant (\(\sec x\)) is the reciprocal of cosine, i.e., \(\frac{1}{\cos x}\).

Understanding these functions and their reciprocal relationships is essential for solving and simplifying trigonometric identities on your homework.

Simplifying expressions

Simplifying expressions is a crucial step in verifying trigonometric identities. The goal is to manipulate the trigonometric functions in expressions to match a simpler or desired form. This often involves substituting known trigonometric relationships and properties, such as reciprocals or Pythagorean identities.
In our exercise, we simplify the complex fraction \(\frac{\tan x}{\sec x}\) by substituting \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec x = \frac{1}{\cos x}\).
Once substitution is completed, you transform the division of fractions into multiplication, allowing for cancellation of terms:

• Multiply \(\frac{\sin x}{\cos x}\) by \(\cos x\)
• The cosine terms \(\cos x\) cancel out, leaving \(\sin x\).

Effectively simplifying expressions helps make the resolution clear, proving that both sides of the identity are equal.

Trigonometric identities verification

Trigonometric identities verification is a process that establishes the equality of two trigonometric expressions for all possible values of the variables involved. The goal is to demonstrate through manipulation and simplification that the left-hand side (LHS) of an expression equals the right-hand side (RHS).
To verify the identity \(\frac{\tan x}{\sec x} = \sin x\), we focus on simplifying the LHS. By expressing both \(\tan x\) and \(\sec x\) in terms of \(\sin x\) and \(\cos x\), we can directly simplify the expression down to \(\sin x\)

• Begin by recalling the definitions of \(\tan x\) and \(\sec x\).
• Simplify the complex fraction by performing multiplication.
• Verify that the final simplified form matches the RHS of the identity.

Each proverbial step ensures that the transformation holds universally across all applicable values, making it easier to understand and verify trigonometric identities in your practice.