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Chapter 6: Problem 69

In Exercises \(69-82,\) prove the given identities. $$\cos (x+x)=\cos ^{2} x-\sin ^{2} x$$

Short Answer

Expert verified
The given identity is correct. The proof relies on using the double angle identity for cosine and the Pythagorean Identity. By substituting these identities into the equation, it can be simplified to show that both sides are indeed equal.

Step by step solution

01

Use the Double Angle Identity

The Double Angle Identity for cosine states that \(\cos(2x) = \cos^2 x - \sin^2 x\). So, we can rewrite the left-hand side of the given identity as \(\cos(2x)\). Hence, we have \(\cos(2x)=\cos^2 x - \sin^2 x\).
02

Use the Pythagorean Identity

The Pythagorean Identity states that \(\cos^2 x + \sin^2 x = 1\). Now, the equation is \(\cos(2x)=1 - 2\sin^2 x\). Using the definition of cosine double angle, which is also \(cos(2x) = 2\cos^2 x -1\), we can set these two equal to each other. Hence, we get \(1 - 2\sin^2 x = 2\cos^2 x -1\). These are equivalent, hence the identity is true.
03

confirm your answer

To make sure the given identity is indeed correct, plug some known values for \(x\) into both sides of the equations (famous angles are often used for this, such as \(0,\frac{\pi}{4},\frac{\pi}{2},\pi,\frac{3\pi}{2},2\pi\) ), and confirm that both sides compute to the same value. Factors of \(\pi\) are typically used since they simplify trigonometric calculations and tend to be the points around the unit circle.

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Key Concepts

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

Double Angle Identity
Understanding the double angle identity is crucial when working with trigonometric functions. It expresses a cosine of twice an angle in terms of the squares of cosine and sine of the original angle. Specifically, the identity states that \( \cos(2x) = \cos^2 x - \sin^2 x \).

This identity allows for the simplification of trigonometric expressions and is particularly useful in proving other trigonometric identities. In the context of the given exercise, we apply this identity to show that the cosine of a sum of the same angle (\(x + x\)) is equivalent to the expression \( \cos^2 x - \sin^2 x \). This transformation is the first step in a multi-step process to confirm the identity.
Pythagorean Identity
The Pythagorean identity is a fundamental relationship in trigonometry that connects the sine and cosine functions of the same angle. It is given by the equation \( \cos^2 x + \sin^2 x = 1\), which is derived from the Pythagorean theorem by considering a unit circle.

This identity tells us that for any angle \(x\), the square of its sine plus the square of its cosine will always equal one. It's a useful tool for simplifying complex trigonometric expressions. In our exercise, we use it to transition between expressions involving solely \(\cos(2x)\) and those involving \(\sin^2 x\) or \(\cos^2 x\). This step is essential for demonstrating the equivalence requested in the identity.
Cosine Function
The cosine function, represented as \(\cos\), is one of the primary trigonometric functions. It relates the angle of a right-angled triangle to the ratio of the adjacent side over the hypotenuse in the triangle. In the context of the unit circle, it gives the \(x\)-coordinate of a point on the circle corresponding to a given angle.

In trigonometric identities, particularly those involving angles and their multiples, the cosine function plays a significant role. It has several properties, such as even symmetry and periodicity, which are leveraged when verifying identities. In the given exercise, the cosine function's properties are utilized through its double angle identity to show how \(\cos(x + x)\) can be transformed into an expression using only powers of \(\cos\) and \(\sin\).

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Most popular questions from this chapter

An object is thrown upward at an acute angle \(\theta\) with respect to the horizontal. The object leaves the thrower's hand at a speed of 64 feet per second. Its horizontal coordinate \(x\) is given by \(x=(64 \cos \theta) t,\) where \(x\) is in feet and \(t\) is the time (in seconds) since the object was thrown. If \(x=64 \sqrt{3}\) when \(t=2 \sqrt{3},\) and the expression for \(x\) can be written as \(x=(64 \sin (\theta-c)) t\) for some \(c\) that \(0^{\circ} < \theta-c < 90^{\circ},\) find the values of \(\theta\) and \(c\) (in degrees).

In Exercises \(63-68,\) write in terms of a single trigonometric function of just \(x\). $$\cos \left(x+\frac{3 \pi}{2}\right)$$

In Exercises \(63-68,\) write in terms of a single trigonometric function of just \(x\). $$\sin (x+n \pi), n \text { odd }$$

In Exercises \(69-82,\) prove the given identities. $$\tan (\pi-x)=-\tan x$$

In Exercises \(69-82,\) prove the given identities. $$\sin \left(x-\frac{\pi}{2}\right)=-\sin \left(\frac{\pi}{2}-x\right)$$
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