Problem 73 - Algebra and Trigonometry

Question

\(73-90\) Prove the identity. $$ \cos ^{2} 5 x-\sin ^{2} 5 x=\cos 10 x $$

Step-by-Step Solution

Answer

The identity is true by directly applying the double angle formula for cosine.

Step 1: Recall the Double Angle Formula

The double angle formula for cosine can be given by:\[ \cos(2x) = \cos^2(x) - \sin^2(x) \]Note that the expression on the left-hand side of our problem is \(\cos^2(5x) - \sin^2(5x)\), which matches the double angle identity where \(x = 5x\).

Step 2: Apply the Double Angle Formula

We can directly apply the double angle formula to our given identity:\[ \cos^2(5x) - \sin^2(5x) = \cos(2 \times 5x)\]This simplifies to:\[ \cos(10x)\]Which is exactly the right-hand side of the equation we set out to prove.

Key Concepts

Double Angle Formulas

Double angle formulas are fundamental in trigonometry and are used to simplify expressions in functions like cosine and sine. They transform expressions involving trigonometric functions of angles to expressions involving different angles, usually simpler in form or related to double the original angle. The most common double angle formulas are:

• For cosine: \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)
• For sine: \( \sin(2\theta) = 2\sin(\theta) \cos(\theta) \)

These formulas can also be rearranged using the Pythagorean identities \( \cos^2(\theta) + \sin^2(\theta) = 1 \) to express each function in terms of one trigonometric function only.Understanding and memorizing these formulas enables you to manipulate trigonometric expressions more effectively, allowing you to simplify complex trigonometric identities or solve trigonometric equations by reducing them to simpler, equivalent forms.

Cosine Function

The cosine function, denoted as \( \cos \), is a fundamental trigonometric function that represents the x-coordinate of a point on a unit circle. The angle is measured from the positive x-axis in a counter-clockwise direction. When you encounter expressions like \( \cos^2(x) - \sin^2(x) \), they can often be related to double angle formulas, as seen in the exercise where this expression represents \( \cos(2x) \).Here are some key attributes of the cosine function:

• It has a range of \([-1, 1]\).
• The cosine function is even, meaning \( \cos(-x) = \cos(x) \).
• The cosine of 0 or any multiple of \(2\pi\) is 1.

The cosine function is extensively used in various fields such as physics, engineering, and computer science, particularly in wave and oscillation modeling, thanks to its periodic nature.

Sine Function

The sine function, represented by \( \sin \), is another cornerstone of trigonometry. It relates the angle of a point on the unit circle to its y-coordinate. Like the cosine function, it is periodic and fundamental for understanding oscillatory motion and wave functions.For relationships involving the sine function, the double angle formula \( \sin(2x) = 2\sin(x)\cos(x) \) is particularly useful for simplifying or transforming expressions. Primary characteristics of the sine function include:

• The range is \([-1, 1]\), just like the cosine function.
• The sine function is odd, meaning \( \sin(-x) = -\sin(x) \).
• Sine of 0 is 0, aligning with the y-coordinate at the origin of a unit circle.

Grasping the behavior of the sine function is crucial in subjects like physics and engineering, as it forms the basis for understanding wave phenomena, oscillations, and alternating currents in circuits.