Problem 15 Prove the identity. $$\frac{\t... [FREE SOLUTION]

Chapter 7: Problem 15

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

Short Answer

Expert verified

Question: Prove the identity $\frac{\tan x}{\sec x}=\sin x$ Solution: To prove the given identity, follow these steps: Step 1: Rewrite the expressions in terms of sine and cosine. Replace $\tan x$ with $\frac{\sin x}{\cos x}$ and $\sec x$ with $\frac{1}{\cos x}$. Step 2: Substitute the expressions into the identity. The identity becomes: $$\frac{\frac{\sin x}{\cos x}}{\frac{1}{\cos x}}=\sin x$$ Step 3: Simplify the left side of the identity. Rewrite the expression as a product: $$\frac{\sin x}{\cos x} \cdot \frac{\cos x}{1} =\sin x$$ The $\cos x$ in the numerator and denominator cancel out, leaving: $$\sin x=\sin x$$ Thus, the identity is proven.

Step by step solution

Rewrite the expressions in terms of sine and cosine.

We will rewrite the expression $\frac{\tan x}{\sec x}$ in terms of sine and cosine. The tangent function, $\tan x$, can be expressed as $\frac{\sin x}{\cos x}$, while the secant function, $\sec x$, can be expressed as $\frac{1}{\cos x}$.

Substitute the expressions into the identity

By substituting the expressions from step 1, our identity now becomes: $$\frac{\frac{\sin x}{\cos x}}{\frac{1}{\cos x}}=\sin x$$

Simplify the left side of the identity

Now, we will simplify the left side of the equation. To do this, we can rewrite the expression as a product: $$\frac{\sin x}{\cos x} \cdot \frac{\cos x}{1} =\sin x$$ Notice that the $\cos x$ in the numerator and denominator will cancel out: $$\frac{\sin x}{\cancel{\cos x}} \cdot \frac{\cancel{\cos x}}{1} =\sin x$$ Leaving us with: $$\sin x=\sin x$$ The identity has been proven.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Function

The tangent function, often denoted as $ \tan x $, is a fundamental trigonometric function. It is defined using the sine and cosine functions. The relationship can be expressed as:\[ \tan x = \frac{\sin x}{\cos x} \]This equation illustrates that the tangent function is the ratio of the sine of angle $ x $ to the cosine of angle $ x $. It plays a crucial role in trigonometry, helping to solve right-angled triangle problems and analyze periodic behavior.

The tangent function is periodic with a period of $ \pi $ radians.
It is important to note that $ \tan x $ is undefined when $ \cos x = 0 $, which occurs at $ x = \frac{\pi}{2} + n\pi $, where $ n $ is an integer.
This function is positive in the first and third quadrants of the unit circle, and negative in the second and fourth.

The relationship $ \tan x = \frac{\sin x}{\cos x} $ is especially useful when proving identities or simplifying trigonometric expressions.

Secant Function

The secant function, denoted $ \sec x $, is another critical trigonometric function. It is the reciprocal of the cosine function. The definition is as follows:\[ \sec x = \frac{1}{\cos x} \]This implies that the secant function is undefined where the cosine function equals zero. These points occur at $ x = \frac{\pi}{2} + n\pi $, where $ n $ is any integer.

Since $ \sec x $ is the reciprocal of $ \cos x $, it inherits the properties related to the sign of $ \cos x $: positive in the first and fourth quadrants, negative in the second and third.
The secant function has a period of $ 2\pi $, like the cosine function.
Its role is vital when working with trigonometric identities, especially those requiring transformations or inversions of cosine values.

Because $ \sec x = \frac{1}{\cos x} $, it simplifies greatly into multiplication with $ \tan x $, especially useful in proving the identity given in the exercise.

Sine Function

The sine function, represented as $ \sin x $, is one of the primary trigonometric functions. It is critical in describing wave patterns, circular motion, and oscillations in physics and engineering.

The sine function has a range of values from $-1$ to $1$.
It is periodic with a period of $ 2\pi $, which means it repeats its values every $ 2\pi $ radians.
The function is defined as the y-coordinate of the corresponding angle $ x $ on the unit circle.

The sine function is often used in identity proofs, as seen in the exercise provided. By expressing $ \frac{\tan x}{\sec x} $ in terms of $ \sin x $, we can directly simplify the equation. The sine function's basic identity helps streamline the verification of other complex identities.

91影视

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

Short Answer

Step by step solution

Rewrite the expressions in terms of sine and cosine.

Substitute the expressions into the identity

Simplify the left side of the identity

Key Concepts

Tangent Function

Secant Function

Sine Function

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