Question

Prove each identity. a) \(\frac{\csc x}{2 \cos x}=\csc 2 x\) b) \(\sin x+\cos x \cot x=\csc x\)

Short answer

a) \(\frac{1}{2 \sin x \cos x} = \csc 2 x \) b) \( \frac{1}{\sin x} = \csc x \).

Step-by-step solution

Rewrite \(\frac{\csc x}{2 \cos x}=\csc 2 x \)

Recall that \(\text{csc} x \) is the reciprocal of \(\text{sin} x \) and \(\text{cos} x \) is the cosine function. We'll start with the given equation \(\frac{\csc x}{2 \cos x}=\csc 2 x \) and use trigonometric identities to prove it.

Substitute trigonometric identities

Rewrite \(\text{csc} x \) as \(\frac{1}{\text{sin} x} \): \[ \frac{\frac{1}{\sin x}}{2 \cos x} = \csc 2 x \]

Simplify the left-hand side

Simplify the fraction: \[ \frac{1}{2 \sin x \cos x} = \csc 2 x \]

Use double angle identity for sine

Recall that \(\text{sin} 2x = 2 \sin x \cos x \): \[ \frac{1}{\sin 2x} = \csc 2 x \]

Conclude the proof for part a)

Recognize that \(\frac{1}{\sin 2x} \) is indeed \(\text{csc} 2x \). Therefore: \[ \frac{\csc x}{2 \cos x} = \csc 2 x \]

Rewrite \( \sin x + \cos x \cot x = \csc x \)

Recall that \(\text{cot} x \) is \(\frac{\text{cos} x}{\text{sin} x} \). We'll start with the given equation \( \sin x + \cos x \cot x = \csc x \) and use trigonometric identities to prove it.

Substitute trigonometric identities

Rewrite \(\text{cot} x \) as \(\frac{\cos x}{\sin x} \): \[ \sin x + \cos x \left( \frac{\cos x}{\sin x} \right) = \csc x \]

Simplify the left-hand side

Combine and simplify the terms: \[ \sin x + \frac{\cos^2 x}{\sin x} = \csc x \]

Combine fractions

Express a common denominator: \[ \frac{\sin^2 x + \cos^2 x}{\sin x} = \csc x \]

Use Pythagorean identity

Recall that \(\text{sin}^2 x + \text{cos}^2 x = 1 \): \[ \frac{1}{\sin x} = \csc x \]

Conclude the proof for part b)

Recognize that \( \frac{1}{\sin x} \) is indeed \( \text{csc} x \). Therefore: \[ \sin x + \cos x \cot x = \csc x \]

Key concepts

cosecant function

The cosecant function, denoted as \(\text{csc} x\), is the reciprocal of the sine function. This means that if you know the value of \(\text{sin} x\), you can find \(\text{csc} x\) using the formula: \[\text{csc} x = \frac{1}{\text{sin} x}\]
The cosecant function is useful in various trigonometric identities and equations. For example, in the given exercise, we use the definition of \(\text{csc} x\) to begin simplifying the given expression and eventually prove the identity.

double angle identity

Double angle identities are trigonometric formulas that express functions of twice an angle in terms of functions of the original angle. One key double angle identity for sine is: \[\text{sin} 2x = 2 \text{sin} x \text{cos} x\]
This identity was crucial in solving part (a) of the exercise. By recognizing and applying this identity, we could transform the expression \(\frac{1}{2 \text{sin} x \text{cos} x}\) into \(\frac{1}{\text{sin} 2x}\), which matches the definition of the cosecant function for double angles, \(\text{csc} 2x\).
Many other double angle identities exist, such as those for cosine and tangent, which also play vital roles in simplifying and solving trigonometric equations.

Pythagorean identity

The Pythagorean identity is one of the fundamental identities in trigonometry. It states that: \[\text{sin}^2 x + \text{cos}^2 x = 1\]
This identity is derived from the Pythagorean Theorem and is used frequently to simplify expressions involving sine and cosine.
In part (b) of the given exercise, the Pythagorean identity was used to simplify the fraction \(\frac{\text{sin}^2 x + \text{cos}^2 x}{\text{sin} x}\) into \(\frac{1}{\text{sin} x}\), leading directly to the conclusion \(\text{csc} x\).

cotangent function

The cotangent function, denoted as \(\text{cot} x\), is the reciprocal of the tangent function. It can be expressed as: \[\text{cot} x = \frac{\text{cos} x}{\text{sin} x}\]
In the given exercise, we used this relationship to solve part (b). By replacing \(\text{cot} x\) with \(\frac{\text{cos} x}{\text{sin} x}\), we could simplify the expression \(\text{sin} x + \text{cos} x \text{cot} x\) into a single fraction, which was then further simplified using the Pythagorean identity.
The cotangent function is particularly useful when dealing with identities and equations that involve division of trigonometric functions.