Problem 57 - Algebra and Trigonometry

Question

Verify the identity: $$ \sin 2 x=2 \cot x \sin ^{2} x $$

Step-by-Step Solution

Answer

After all the steps of simplification and applying the necessary trigonometric identities, it is confirmed that \(\sin 2x\) is indeed equal to \(2 \cot x \sin^2 x\). This verifies the identity.

Step 1: Express \(\sin 2x\)

First replace \(\sin 2x\) on the left side of the equation with \(2 \sin x \cos x\). This is possible due to the double angle identity \(\sin 2x = 2 \sin x \cos x\). So the given equation is now represented as: \(2 \sin x \cos x = 2 \cot x \sin^2 x\).

Step 2: Express \(\cot x\)

Now, replace \(\cot x\) on the right side of the equation with \(\cos x / \sin x\). This is possible due to the cotangent identity \(\cot x = \cos x / \sin x\), The given equation is now represented as: \(2 \sin x \cos x = 2 \cdot (\cos x / \sin x) \cdot \sin^2 x\).

Step 3: Simplify the right side of the equation

Multiply out on the right side of the equation to get: \(2 \sin x \cos x = 2 \cos x (\sin x \cdot \sin x)\). Simplify to get the right side to: \(2 \sin x \cos x\).

Step 4: Compare both sides

Now we see that both sides of the equation are equal, hence, confirming the identity which was required to be proved.

Key Concepts

Double Angle Identity

The double angle identity is an important trigonometric formula that helps simplify expressions involving angles. When working with quadratic trigonometric expressions, this identity is particularly useful.

The double angle identity for sine is given by:

• \( \sin 2x = 2 \sin x \cos x \)

At first glance, this identity might look complex, but it is actually just a special case of the sum of angles formula for sine. It tells us how to express the sine of twice an angle in terms of the sine and cosine of the original angle.

Using this identity allows us to break down complex trigonometric functions into simpler components, making it easier to solve problems. In our example, the double angle identity transforms \( \sin 2x \) into \( 2 \sin x \cos x \), which we can work with more easily to verify the given identity.

Cotangent Identity

The cotangent identity is another fundamental trigonometric formula, providing a way to express the cotangent function in terms of sine and cosine. Understanding this identity can help make complex equations simpler.

• \( \cot x = \frac{\cos x}{\sin x} \)

In the context of verifying trigonometric identities, the cotangent identity acts as a bridge between various trigonometric functions.

By substituting this expression into equations, we can reduce the complexity and manage the functions more effectively. In our specific problem, replacing \( \cot x \) with \( \frac{\cos x}{\sin x} \) on the right side of the equation facilitated the simplification process.

This substitution was key in showing that both sides of the given identity could be made equivalent, emphasizing the power of trigonometric identities in problem-solving.

Sine Function

The sine function is one of the primary trigonometric functions used to relate angles to side lengths in right-angled triangles. It is central to many forms of wave-based analysis in both mathematics and physics.

• The basic definition is: \( \sin x = \frac{\text{opposite side}}{\text{hypotenuse}} \)

Beyond right triangles, the sine of an angle is an essential part of periodic functions such as sound and light waves.

In expressions involving angles, the sine function shows how those angles are expressed as ratios. The sine function was a key element in our example as it was manipulated through trigonometric identities to verify the given equation.

Understanding the sine function's properties helps in breaking down and solving trigonometric expressions, such as using identities to simplify and prove equalities.

Cosine Function

The cosine function is another foundational trigonometric function, complementary to the sine function. Understanding how it relates to angles is crucial in trigonometry.

• The basic definition is: \( \cos x = \frac{\text{adjacent side}}{\text{hypotenuse}} \)

Often paired with the sine function, cosine is integral to modeling oscillations and waves, and its calculations appear across various scientific fields.

In the context of trigonometric identities, cosine often works in tandem with sine to simplify expressions. For example, in proving our identity, the cosine function combines with sine through the double angle identity.

Having a deep understanding of the cosine function allows for better manipulation of trigonometric expressions, ultimately making solving such identities more intuitive.