Problem 26 - Algebra 2

Question

Verify that each of the following is an identity. $$ \tan ^{2} \frac{x}{2}=\frac{1-\cos x}{1+\cos x} $$

Step-by-Step Solution

Answer

The equation is indeed an identity.

Step 1: Use Double Angle Formula for Cosine

The double angle formula for cosine is \( \cos x = 1 - 2\sin^2\left(\frac{x}{2}\right) \). Substitute this into the right side of the identity: \( \frac{1 - \cos x}{1 + \cos x} = \frac{1 - (1 - 2\sin^2\left(\frac{x}{2}\right))}{1 + (1 - 2\sin^2\left(\frac{x}{2}\right))}. \)

Step 2: Simplify the Numerator and Denominator

After substitution, simplify the expression: - Numerator: \( 1 - (1 - 2\sin^2(\frac{x}{2})) = 2\sin^2(\frac{x}{2}) \) - Denominator: \( 1 + (1 - 2\sin^2(\frac{x}{2})) = 2 - 2\sin^2(\frac{x}{2}) = 2\cos^2(\frac{x}{2}) \) since \( \cos^2(\frac{x}{2}) = 1 - \sin^2(\frac{x}{2}) \).

Step 3: Simplify the Fraction

Now, simplify the fraction: \[ \frac{2\sin^2(\frac{x}{2})}{2\cos^2(\frac{x}{2})} = \frac{\sin^2(\frac{x}{2})}{\cos^2(\frac{x}{2})} \]. This equals \( \tan^2(\frac{x}{2}) \) by definition of the tangent function.

Step 4: Conclusion

We have shown that \( \frac{1 - \cos x}{1 + \cos x} = \tan^2(\frac{x}{2}) \). Therefore, the given equation is an identity.

Key Concepts

Double Angle Formula

The double angle formula is a mathematical expression used to relate trigonometric functions of angles.It is especially useful in simplifying expressions and verifying identities, like the one in this exercise.For cosine, the double angle formula is given by:\[ \cos x = 1 - 2\sin^2\left(\frac{x}{2}\right) \]This formula comes in handy when we need to express the trigonometric functions involving angle \( x \) in terms of \( \frac{x}{2} \).In our exercise, we use this formula to rewrite \( \cos x \), thereby simplifying the expression \( \frac{1 - \cos x}{1 + \cos x} \) that helps verify the identity.This transformation is crucial in the understanding and simplification of trigonometric identities.

Tangent Function

The tangent function is a fundamental trigonometric function defined as the ratio of sine to cosine:\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]For our exercise, it is particularly helpful to look at \( \tan^2(\frac{x}{2}) \), which is the square of the tangent function at the angle \( \frac{x}{2} \).This is what the identity in question simplifies to.By understanding this, you can readily verify that the rearranged expression derived using the double angle formula equates to the square of the tangent.This shows how different trigonometric expressions can be related using the fundamental definition of the tangent function.

Trigonometric Functions

Trigonometric functions are a group of functions that relate angles of a triangle to the lengths of its sides.They are foundational in understanding many aspects of mathematics and engineering, especially in solving geometric problems.The primary functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)).These functions have relationships and identities, like the one in this exercise:- The identity helps relate different functions through simplifications.- The use of identities, like the double angle formula, aids in transforming complex expressions into something more manageable.Understanding these basic relationships allows you to manipulate and solve trigonometric expressions effectively.

Simplification Process

The simplification process is crucial in confirming trigonometric identities.It involves manipulating expressions to transform one form into another, often more usable form.For our current exercise:- **Step 1**: Use the double angle formula to transform \( \cos x \) into terms of \( \sin(\frac{x}{2}) \). This step is essential as it sets up the substitution needed.- **Step 2**: Simplify the numerator and the denominator separately, which leads to expressions in terms of either \( \sin \) or \( \cos \) of \( \frac{x}{2} \).- **Step 3**: Further simplify the resulting fraction to isolate \( \tan^2(\frac{x}{2}) \), demonstrating the identity.Each step requires attention to detail, understanding how to apply identities, and recognizing common forms to proceed efficiently.