Problem 72 Prove the following identities. ... [FREE SOLUTION]

Chapter 1: Problem 72

Prove the following identities. $$\sec (x+\pi)=-\sec x$$

Short Answer

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Question: Prove that the identity $$\sec (x+\pi)=-\sec x$$ holds true. Solution: 1. Rewrite the sec terms using the cos function: $$\frac{1}{\cos(x+\pi)} = -\frac{1}{\cos x}$$ 2. Simplify the left side using the angle addition formula for cos: $$\cos(x+\pi) = \cos x \cos \pi - \sin x \sin \pi$$ 3. Evaluate the trigonometric functions for $\pi$: $$\cos \pi = -1$$ and $$\sin \pi = 0$$ 4. Substitute these values back into the equation: $$\cos(x+\pi) = -\cos x$$ 5. Rewrite the left side of the identity in terms of sec: $$\frac{1}{\cos(x+\pi)} = \frac{1}{-\cos x}$$ Therefore, we have proven that $$\sec (x+\pi)=-\sec x$$.

Step by step solution

Rewrite sec using cos

To start, let's rewrite the sec terms using cos function, since we know that: $$\sec x = \frac{1}{\cos x}$$. So, the identity becomes: $$\frac{1}{\cos(x+\pi)} = -\frac{1}{\cos x}$$

Simplify the left side using the angle addition formula

The angle addition formula for cos is: $$\cos(x+y) = \cos x \cos y - \sin x \sin y$$. Using this formula, we can simplify $\cos(x+\pi)$ as follows: $$\cos(x+\pi) = \cos x \cos \pi - \sin x \sin \pi$$

Evaluate the trigonometric functions for $\pi$

Evaluate the trig functions for $\pi$: $$\cos \pi = -1$$ $$\sin \pi = 0$$ Now, substitute these values back into the equation from Step 2: $$\cos(x+\pi) = \cos x (-1) - \sin x (0)$$ So, we get: $$\cos(x+\pi) = -\cos x$$

Express the left side of the identity in terms of sec

Now that we have simplified $\cos(x+\pi)$, we can rewrite the left side of the identity in terms of sec: $$\frac{1}{\cos(x+\pi)} = \frac{1}{-\cos x}$$

Compare both sides of the identity

Comparing both sides of the identity, we can see that they are equal: $$\frac{1}{\cos(x+\pi)} = \frac{1}{-\cos x}$$ So, we have proven that $$\sec (x+\pi)=-\sec x$$

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Key Concepts

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

Secant Function

The Secant function, often abbreviated as "sec," is one of the six primary trigonometric functions. It's the reciprocal of the cosine function. Thus, you can express the secant of an angle $x$ as:
\[\sec x = \frac{1}{\cos x}\]This means that secant tells us to "flip" the cosine value of an angle. So, if you know $\cos x$, you can easily find $\sec x$ by taking its reciprocal.
The secant function has specific characteristic properties:

It is undefined when $\cos x = 0$, because division by zero is not possible.
The graph of the secant function has vertical asymptotes where the cosine function crosses zero.
It is an even function, which means $\sec(-x) = \sec x$.

Understanding the secant function is essential in trigonometry, especially in solving identities and equations involving angles.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, usually represented as $\cos$. It measures the horizontal position or "adjacency" in a right triangle with respect to an angle. In the unit circle, it represents the x-coordinate of a point.
The formula for cosine in terms of angle $x$ is defined as:
\[\cos x = \frac{adjacent}{hypotenuse}\]In the context of the unit circle, the cosine value is simply the x-coordinate of the corresponding point on the circle.
Cosine function characteristics include:

It is a periodic function with a period of $2\pi$.
It is an even function, so $\cos (-x) = \cos x$.
The cosine of $0$ is $1$ and the cosine of $\pi$ is $-1$.

These properties make the cosine function essential for various applications in physics, engineering, and signal processing.

Angle Addition Formula

The angle addition formula for trigonometric functions helps you find the cosine or sine of the sum of two angles. For cosine, it looks like this:
\[\cos(x + y) = \cos x \cos y - \sin x \sin y\]This indicates that the cosine of the sum of two angles involves both sine and cosine components of each angle.
This formula is vital when simplifying expressions or solving problems that involve angles that are not discrete or standard values from trigonometric tables.

Using the angle addition formula can help transform and simplify trigonometric expressions.
It's extremely useful in proving identities, such as the one in our problem: $\sec(x + \pi) = -\sec x$.
It is primarily used to break down complex trigonometric expressions into simpler forms.

Understanding and applying the angle addition formula allows you to expand your ability to work with more complex trigonometric functions and equations.

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91影视

Prove the following identities. $$\sec (x+\pi)=-\sec x$$

Short Answer

Step by step solution

Rewrite sec using cos

Simplify the left side using the angle addition formula

Evaluate the trigonometric functions for \(\pi\)

Express the left side of the identity in terms of sec

Compare both sides of the identity

Key Concepts

Secant Function

Cosine Function

Angle Addition Formula

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