Problem 51 - Algebra and Trigonometry

Question

Verify each identity. \(\cos ^{4} t-\sin ^{4} t=1-2 \sin ^{2} t\)

Step-by-Step Solution

Answer

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

Step 1: Recognizing the Expression as a Difference of Squares

The given expression \(\cos ^{4} t-\sin ^{4} t\) is a difference of two squares. We can write \(\cos^4(t) - \sin^4(t) = (\cos^2(t))^2 - (\sin^2(t))^2\), which can be factored into \((\cos^2(t) - \sin^2(t))(\cos^2(t) + \sin^2(t))\). We know that by Pythagorean identity, \(\cos^2(t) + \sin^2(t) = 1\), so the expression simplifies to \(\cos^2(t) - \sin^2(t)\)

Step 2: Apply Double-Angle Identity

We can express \(\cos^2(t) - \sin^2(t)\) in terms of double-angle using the identity \(\cos(2t) = \cos^2(t) - \sin^2(t)\). So the expression becomes \(\cos(2t)\)

Step 3: Simplify \(\cos(2t)\) to match the right side of the equation

We can convert the \(\cos(2t)\) back into sine using the identity \(\cos(2t) = 1 - 2\sin^2(t)\). So the expression becomes \(1-2 \sin ^{2} t\)

Step 4: Completing the Verification

We have transformed the left side of the equation into the same form as the right side, thus verifying the identity \(\cos ^{4} t-\sin ^{4} t=1-2 \sin ^{2} t\).

Key Concepts

Difference of Squares

In mathematics, the term "difference of squares" refers to an expression that can be written in the form of \(a^2 - b^2\). This is a crucial algebraic identity because it can be factored into \((a - b)(a + b)\). This simplification makes complex expressions more manageable and is often the first step in solving equations.
For instance, in our original exercise with the expression \(\cos^4(t) - \sin^4(t)\), we identify it as a difference of squares by recognizing that each term is a square: \((\cos^2(t))^2 - (\sin^2(t))^2\).
To simplify this, apply the difference of squares identity:

• Factor it as \((\cos^2(t) - \sin^2(t))(\cos^2(t) + \sin^2(t))\).
• You need to remember that \(\cos^2(t) + \sin^2(t)\) equals 1.
• This reduces the whole expression to \(\cos^2(t) - \sin^2(t)\).

This is a clear demonstration of how utilizing the difference of squares can directly simplify and verify expressions in trigonometry.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry. It states that \(\sin^2(\theta) + \cos^2(\theta) = 1\). It's derived from the Pythagorean theorem applied to a unit circle.
In the context of the original exercise, this identity allows us to rewrite and simplify trigonometric expressions effectively. For example, in the expression \((\cos^2(t) + \sin^2(t))\), which arises from the difference of squares factoring, we can directly substitute it with 1:

• This substitution simplifies the expression further, as seen when \(\cos^2(t) + \sin^2(t) = 1\).
• Reducing complex expressions becomes easier, enabling you always to rely on this constant relationship.

The Pythagorean identity not only makes calculations simpler but is fundamental in verifying identities in various trigonometric problems, such as the one presented.

Double-Angle Identity

Double-angle identities are fascinating and allow trigonometric expressions to be expressed more concisely. The double-angle identity for cosine states \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\). This can also be written in other equivalent forms, such as \(1 - 2\sin^2(\theta)\).
In our exercise, after using the difference of squares and simplifying with the Pythagorean identity, we arrived at \(\cos^2(t) - \sin^2(t)\).

• By recognizing this expression as the double angle identity, it transforms directly into \(\cos(2t)\).
• Further, to match the right side of the equation, another form of the double-angle identity is used: \(1 - 2\sin^2(t)\).

This simple transition shows the power of double-angle identities, turning seemingly complex trigonometric expressions into something more tractable. Using these identities strategically can simplify and solve a variety of mathematical problems effortlessly.