Problem 40 - Algebra and Trigonometry

Question

Verify each identity. \(\cot ^{2} 2 x+\cos ^{2} 2 x+\sin ^{2} 2 x=\csc ^{2} 2 x\)

Step-by-Step Solution

Answer

The given trigonometric identity is verified and hence is correct.

Step 1: Recall and apply trigonometric identities

Start by recalling the identity \(\cot ^{2}x = \csc^{2}x - 1\), and apply this to \(\cot ^{2} 2x\) on the left side. That gives \(\csc ^{2} 2x - 1 + \cos ^{2} 2x + \sin ^{2} 2x\).

Step 2: Simplify using fundamental trigonometric identity

Next, recognize that \(\cos ^{2} 2x + \sin ^{2} 2x\) is one of the fundamental identities in trigonometry, which simplifies to 1. So the left hand side becomes: \(\csc ^{2} 2x - 1 + 1 = \csc ^{2} 2x\).

Step 3: Confirm the equation

Now confirm that both sides of the original equation are the same, that is: \(\cot ^{2} 2 x+\cos ^{2} 2 x+\sin ^{2} 2 x = \csc ^{2} 2 x\) indeed.

Key Concepts

Cotangent Identity

The cotangent identity is an important aspect of trigonometry. It ties together various trigonometric functions elegantly. The identity is expressed as: \[\cot ^{2}x = \csc^{2}x - 1\]This identity is essential because it shows the relationship between cotangent and cosecant, two primary trigonometric functions.
Working with such identities allows us to simplify and verify more complex equations efficiently.

• The \(\cot\) function is the reciprocal of \(\tan\), so \(\cot x = \frac{1}{\tan x}\).
• The \(\csc\) function is the reciprocal of \(\sin\), so \(\csc x = \frac{1}{\sin x}\).

By utilizing the identity, we can transform and work through trigonometric expressions more easily, as seen in the solution process of the given exercise.

Pythagorean Identities

The Pythagorean identities are among the most fundamental tools in trigonometry. These identities stem from the Pythagorean theorem and form the backbone of many trigonometric simplifications. The most commonly used Pythagorean identity is:\[\sin ^{2}x + \cos ^{2}x = 1\]In the context of this exercise, another variation, derived from this primary identity, is utilized:

• \(\cos ^{2} 2x + \sin ^{2} 2x = 1\)

This tells us that for any angle \(x\), the sum of the squares of the sine and cosine will always equate to 1.
This identity holds true across all angles and is critical for simplifying trigonometric equations. Using this, we can combine different trigonometric terms and simplify complex expressions effectively, as demonstrated in step 2 of the original solution.

Cosecant Identity

The cosecant identity is tightly linked with both sine and the Pythagorean identities. Cosecant, represented as \(\csc\), is the reciprocal of sine, expressed as:\[\csc x = \frac{1}{\sin x}\]The identity used in the verification exercise connections cosecant with cotangent through:

• \(\csc^{2}x = \cot^{2}x + 1\)

By substituting and rearranging identities involving \(\csc\), we can ascertain the validity of trigonometric equations. This identity is particularly useful when verifying more complex statements, as it helps to connect and transform components into simpler forms. By understanding and using the cosecant identity, students can handle various trigonometric identities with greater confidence and accuracy.