Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity. $$ 2 \sin ^{2} 2 t+\cos 4 t=1 $$

Step-by-Step Solution

Answer

The identity is verified: \(2 \sin^{2} 2t + \cos 4t = 1\).

Step 1: Recall the Double Angle Identities

One of the commonly used trigonometric identities for the double angles is \[ \cos(2x) = 1 - 2\sin^2(x) \] and another is \[ \cos(2x) = 2\cos^2(x) - 1 \]. These identities will be helpful in transforming the given expression to verify the identity.

Step 2: Apply the Double Angle Identity to \(\cos(4t)\)

Notice that \(\cos(4t)\) can be expressed using the double angle formula: \[ \cos(4t) = 1 - 2\sin^2(2t). \] Replace \(\cos(4t)\) in the given expression with this identity.

Step 3: Substitute and Simplify the Given Expression

Substitute \(\cos(4t) = 1 - 2\sin^2(2t)\) into the original expression, leading to: \[ 2\sin^2(2t) + (1 - 2\sin^2(2t)) = 1. \] Simplify the expression: \[ 2\sin^2(2t) + 1 - 2\sin^2(2t) = 1. \] The middle terms \(2\sin^2(2t)\) and \(-2\sin^2(2t)\) cancel out, reducing the expression to \[ 1 = 1. \]

Step 4: Confirm the Identity

The simplification results in a true statement, \(1 = 1\), which confirms that the original identity \(2 \sin^{2} 2t + \cos 4t = 1\) is indeed correct. There are no contradictions, confirming the identity.

Key Concepts

Double Angle Formulas

In trigonometry, double angle formulas are essential tools, especially when working with identities. These formulas allow you to express trigonometric functions of double angles (like \(2x\) or \(4t\)) in terms of single angles. A common double angle formula for cosine is:
\[ \cos(2x) = 1 - 2\sin^2(x) \]
This formula highlights how we can represent a trigonometric function using a squared sine function. Another variant involves the cosine function:
\[ \cos(2x) = 2\cos^2(x) - 1 \]
Using these identities makes transforming expressions, like the one in the exercise, possible. We substituted \(\cos(4t)\) with one of the double angle formulas to transform the original equation. This process simplifies the complexity and aids in verifying the identity. Understanding these substitutions improves algebraic manipulation skills in trigonometry.

Sin Squared Identities

Sin squared identities, such as \(\sin^2(x)\) and its relationships in an equation, are fundamental in simplifying expressions. When you encounter an expression like \(2\sin^2(2t)\), it is beneficial to know that through trigonometric identities, it can be broken down or related to other parts of the identity.
For instance, because \(2\sin^2(x) = 1 - \cos(2x)\), the expressions that seem complex can be transformed into more straightforward forms that are easier to verify or calculate.
These identities demonstrate the interconnectedness of trigonometric functions, enriching your understanding of how sine and cosine relate to each other in different contexts. Through substitution and simplification, such as in the verification process, these identities play a pivotal role in managing trigonometric equations.

Cosine Identities

Cosine identities are central in trigonometry and encompass various expressions and transformations involving the cosine function.

• The double-angle identities mentioned before are specific types of cosine identities.
• Basic cosine identities, like \(\cos^2(x) = \frac{1 + \cos(2x)}{2}\), allow for necessary substitutions in equations.

Such expressions offer alternative ways to represent equations, helping to simplify complicated expressions. In our exercise, by expressing \(\cos(4t)\) using a double angle identity, we could transform the original equation into a simpler format. These identities all form an essential toolkit for anyone trying to tackle advanced trigonometric problems or verify complex identities.
By understanding and applying these, one can significantly simplify the calculations or verification processes for different equations, showcasing the power and practicality of cosine identities in trigonometry.