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Chapter 7: Problem 72

Verify that each trigonometric equation is an identity. $$\left(1-\cos ^{2} \alpha\right)\left(1+\cos ^{2} \alpha\right)=2 \sin ^{2} \alpha-\sin ^{4} \alpha$$

Short Answer

Expert verified
The given equation is a trigonometric identity as both sides simplify to equal expressions.

Step by step solution

01

Use Basic Trigonometric Identities

Recall the Pythagorean identity: \( \boxed{\text{\textbf{sin}} ^2 \theta + \text{\textbf{cos}} ^2 \theta = 1} \). Therefore, \( \boxed{\text{\textbf{sin}} ^2 \theta = 1 - \text{\textbf{cos}} ^2 \theta} \). Use this identity to simplify the given equation.
02

Substitute Identity into Equation

Replace \( 1 - \text{\textbf{cos}} ^2 \theta \) with \( \text{\textbf{sin}} ^2 \theta \) in the given equation: \( (1- \text{\textbf{cos}} ^2 \theta )(1+ \text{\textbf{cos}} ^2 \theta ) = 2 \text{\textbf{sin}} ^2 \theta - \text{\textbf{sin}} ^4 \theta \). This transforms into: \( ( \text{\textbf{sin}} ^2 \theta )(1 + \text{\textbf{cos}} ^2 \theta ) \).
03

Expand and Simplify the Left-Hand Side (LHS)

Expand the left-hand side of the equation: \( \text{\textbf{sin}} ^2 \theta + \text{\textbf{sin}} ^2 \theta \text{\textbf{cos}} ^2 \theta \).
04

Factor the Right-Hand Side (RHS)

Factor the right-hand side of the equation: \( 2 \text{\textbf{sin}} ^2 \theta - \text{\textbf{sin}} ^4 \theta \). Observe that this can be written as: \( \text{\textbf{sin}} ^2 \theta (2 - \text{\textbf{sin}} ^2 \theta ) \).
05

Compare Both Sides

Compare the expanded left-hand side with the factored right-hand side of the equation: Both sides should simplify to \( \text{\textbf{sin}} ^2 \theta + \text{\textbf{sin}} ^2 \theta \text{\textbf{cos}} ^2 \theta \). Confirm that \( \text{\textbf{sin}} ^2 \theta (1 + \text{\textbf{cos}} ^2 \theta ) = 2 \text{\textbf{sin}} ^2 \theta - \text{\textbf{sin}} ^4 \theta \). Therefore, the given trigonometric equation is an identity.

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Key Concepts

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

Pythagorean Identity
One of the fundamental trigonometric identities is the Pythagorean identity. It states that for any angle \( \theta \, \ ( \sin^2 \theta + \cos^2 \theta = 1 \). This relationship arises from the Pythagorean theorem applied to a right-angled triangle. In simpler terms, if you draw a right triangle inside a unit circle, the sum of the squares of the opposite side (sine) and the adjacent side (cosine) equals one. This identity helps in transforming and simplifying trigonometric expressions. For instance, we can rearrange it to find expressions for \( \sin^2 \theta \ and \cos^2 \theta, such as \ ( \sin^2 \theta = 1 - \cos^2 \theta \). This rearrangement is crucial in solving many trigonometric equations, just as in the given exercise.
Trigonometric Simplification
Simplifying trigonometric expressions often involves using known identities to replace complex parts with simpler forms. For example, in our exercise, we used the Pythagorean identity to replace \( \1 - \cos^2 \alpha \) with \ \sin^2 \alpha. This kind of substitution makes it easier to manage and transform the expression. When simplifying:
  • Look for known identities.
  • Substitute complex parts with simpler equivalents.
  • Expand and combine like terms if necessary.
  • Factorize expressions to prepare for further simplifications or verifications.
Through these steps, trigonometric identities become manageable and less intimidating to work with.
Verification of Identities
Verifying trigonometric identities means proving that two sides of an equation are equivalent using known identities and algebraic manipulations. In the provided exercise, we verified the identity step-by-step:
  • First, identify the identity to apply (Pythagorean identity).
  • Make substitutions to simplify one or both sides of the equation.
  • Expand, combine like terms, and**: align forms on both sides.
  • Conclude equivalence if both sides match perfectly after simplification.
This methodical approach ensures correctness and helps in solidifying one's understanding of trigonometric identities.

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Most popular questions from this chapter

Verify that each equation is an identity. $$\cot ^{2} \frac{x}{2}=\frac{(1+\cos x)^{2}}{\sin ^{2} x}$$

(This discussion applies to functions of both angles and real numbers.) Consider the following. $$\begin{aligned}\cos (&\left.180^{\circ}-\theta\right) \\\&=\cos 180^{\circ} \cos \theta+\sin 180^{\circ} \sin \theta \\\&=(-1) \cos \theta+(0) \sin \theta \\\&=-\cos \theta\end{aligned}$$ \(\cos \left(180^{\circ}-\theta\right)=-\cos \theta\) is an example of a reduction formula, which is an identity that reduces a function of a quadrantal angle plus or minus \(\theta\) to a function of \(\theta\) alone. Another example of a reduction formula is \(\cos \left(270^{\circ}+\theta\right)=\sin \theta\) Here is an interesting method for quickly determining a reduction formula for a trigonometric function \(f\) of the form \(f(Q \pm \theta),\) where \(Q\) is a quadrantal angle. There are two cases to consider, and in each case, think of \(\boldsymbol{\theta}\) as a small positive angle in order to determine the quadrant in which \(Q \pm \theta\) will lie. Case 1 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(x\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that \(\operatorname{sign}, f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Case 2 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(y\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that sign, the cofunction of \(f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Use these ideas to write reduction formulas for each of the following. $$\sin \left(180^{\circ}+\theta\right)$$

Verify that each equation is an identity. $$\sin 4 x=4 \sin x \cos x \cos 2 x$$

Verify that each equation is an identity. $$\tan 8 \theta-\tan 8 \theta \tan ^{2} 4 \theta=2 \tan 4 \theta$$

Verify that each trigonometric equation is an identity. $$(1+\sin x+\cos x)^{2}=2(1+\sin x)(1+\cos x)$$
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