Problem 30 - Algebra and Trigonometry

Question

Verify each identity. \(1-\frac{\cos ^{2} x}{1+\sin x}=\sin x\)

Step-by-Step Solution

Answer

The identity \(1-\frac{\cos ^{2} x}{1+\sin x}=\sin x\) is verified after we use the Pythagorean trigonometric identity to simplify the left side of the equation and showed it equals \(\sin x\).

Step 1: Use Pythagorean Trigonometric Identity

Apply the Pythagorean identity \(1 - \cos^{2}(x) = \sin^{2}(x)\) on the left side of the expression, thus it becomes \(1-\frac{\sin^{2}x}{1+\sin x}\).

Step 2: Distribute denominator into numerator

Distribute the term in the denominator, \(1 + \sin x \), into the numerator, \(\sin^{2} x\). This becomes \(1 - \sin x - \sin^{3}x\).

Step 3: Use Pythagorean trigonometric identity again

Again, apply the Pythagorean identity, this time replacing \(\sin^{3}x\) with \((1-\cos^{2}x)((1-\cos x)\). The formula becomes \(1 - \sin x - (1 - \cos^{2}x)(1 - \cos x)\).

Step 4: Apply the distributive property

Apply the distributive property in the bracket to get \(1 - \sin x - 1 + \cos x + \cos^{2}x\).

Step 5: Simplify the expression

Simplify the expression to find that \(1 - \sin x - 1 + \cos x + \cos^{2}x = \sin x\)

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. It helps simplify trigonometric expressions and verify identities with sine and cosine functions. The identity is expressed as:

• \( \sin^2(x) + \cos^2(x) = 1 \)

This equation is derived from the Pythagorean theorem applied to the unit circle. Here, the hypotenuse is the radius which is 1, and the opposite and adjacent sides are represented by \( \sin(x) \) and \( \cos(x) \) respectively.
In our example, the Pythagorean identity is used twice:

• First, to replace \( 1 - \cos^2(x) \) with \( \sin^2(x) \). This simplifies the expression and is a crucial first step in solving the problem.
• Later, it's used to transform \( \sin^3(x) \) as needed for further simplification.

By understanding how to apply this identity, you can tackle numerous trigonometric expressions more intuitively.

Simplifying Expressions

Simplifying expressions in trigonometry often involves breaking down complex parts into simpler components. This makes it easier to understand and solve the problem. Here’s how you can think about simplifying:

• Look for identities or properties like the Pythagorean identity that can change the form of the expression. These identities often reveal simpler forms.
• Combine like terms, and remove unnecessary elements.
• In this problem, after applying the Pythagorean identity, simplifying involves reducing the expression by canceling out terms and algebraic simplification.

Each simplification step makes the expression less complex and moves it closer to the goal. Practicing these simplifications can improve your problem-solving skills significantly in trigonometry.

Distributive Property

The distributive property is a foundational algebraic technique used to multiply a single term by two or more terms inside a bracket.Here's how the distributive property works:

• For an expression \( a(b + c) \), it expands to \( ab + ac \).

In our trigonometric problem, the distributive property is crucial when multiplying the denominator \( 1 + \sin(x) \) into the terms like \( \sin^2(x) \) or when dealing with products like \( (1 - \cos^2(x))(1 - \cos(x)) \).
Applying the distributive property simplifies the terms:

• By dispersing the \( \cos \) terms properly, you simplify the expression.
• This step prepares the expression for further reduction by combining similar terms or applying additional identities.

Understanding this property allows you to manage complex expressions and is especially useful in trigonometry and algebra alike.