Problem 75 - Algebra and Trigonometry

Question

\(73-90\) Prove the identity. $$ (\sin x+\cos x)^{2}=1+\sin 2 x $$

Step-by-Step Solution

Answer

The identity \((\sin x+\cos x)^2=1+\sin 2x\) is proven by expanding and using trigonometric identities.

Step 1: Expand the Left Hand Side (LHS)

Start by expanding the expression on the left side of the identity \((\sin x + \cos x)^2\) using the distributive property. The expression becomes \(\sin^2 x + 2\sin x \cos x + \cos^2 x\).

Step 2: Simplify the Trigonometric Functions

Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to simplify the terms. Thus, the LHS becomes \(1 + 2 \sin x \cos x\).

Step 3: Simplify using Double Angle Formula

Recognize that the term \(2 \sin x \cos x\) is equivalent to \(\sin 2x\) by the double angle identity for sine. Therefore, \(1 + 2 \sin x \cos x = 1 + \sin 2x\).

Step 4: Compare Both Sides

Now compare the simplified left side \(1 + \sin 2x\) with the right side of the original identity. Both sides are equal, thus proving the identity.

Key Concepts

Pythagorean Identity

In trigonometry, the Pythagorean Identity is a fundamental relationship that connects the squares of the sine and cosine functions. It is expressed as:\[\sin^2 x + \cos^2 x = 1\].This identity is indispensable when proving various trigonometric equations and simplifying expressions. It tells us that for any angle \(x\), the sum of the squares of \(\sin x\) and \(\cos x\) will always equal one.
When using this identity in problem-solving, it allows us to replace a pair of squared functions with a constant — making equations simpler and often more manageable. In the given exercise, leveraging the Pythagorean Identity helps simplify the left-hand side (LHS) of the equation from \(\sin^2 x + \cos^2 x + 2\sin x\cos x\) to \(1 + 2\sin x \cos x\).
This simplification is crucial for matching the forms on both sides of the identity and proving equivalence.

Double Angle Formula

The Double Angle Formula for sine provides a way to express trigonometric functions involving twice an angle in terms of functions of the original angle. The formula for sine is given by:\[\sin 2x = 2 \sin x \cos x\].This formula emerges from the sum of angles identities and is very handy in solving equations where an angle is doubled. In our exercise, recognizing that the term \(2 \sin x \cos x\) from the LHS can be replaced with \(\sin 2x\) using the double angle formula was a critical step.
By substituting this in the expanded expression, we simplified \(1 + 2 \sin x \cos x\) to \(1 + \sin 2x\).
The simplicity and elegance of the Double Angle Formula not only streamline solving identities but also help reveal deeper connections between trigonometric angles.

Algebraic Expansion

Algebraic expansion involves multiplying expressions out to remove parentheses and produce a simplified form. It's a fundamental technique used in mathematics for simplifying trigonometric expressions and other polynomial expansions.
In the context of trigonometric identities, expanding expressions such as \((\sin x + \cos x)^2\) involves using the distributive property or FOIL (First, Outer, Inner, Last) method. For instance, expanding the exercise’s LHS, we have:\[(\sin x + \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x\].This step is essential, as it sets the stage for leveraging identities like the Pythagorean Identity and Double Angle Formula.
Algebraic expansion simplifies the expressions into a form where specific trigonometric identities can be seamlessly applied, thus ensuring a successful proof of the original identity.