Question

Verify the following are identities. (a) \(\frac{\sin u}{\csc u}+\frac{\cos u}{\sec u}=1\) (b) \(\left(1-\cos ^{2} x\right)\left(1+\cot ^{2} x\right)=1\) (c) \(\sin t(\csc t-\sin t)=\cos ^{2} t\) (d) \(\frac{1-\csc ^{2} t}{\csc ^{2} t}=\frac{-1}{\sec ^{2} t}\)

Short answer

All given expressions are identities: (a) uses Pythagorean identity, (b) simplifies to 1 using trigonometric identities, (c) simplifies to \( \cos^2 t \), and (d) uses Pythagorean identity and algebra.

Step-by-step solution

Simplify Left Side of Identity (a)

Rewrite \( \csc u \) and \( \sec u \) using basic trigonometric identities. \( \csc u = \frac{1}{\sin u} \) and \( \sec u = \frac{1}{\cos u} \). Therefore, \( \frac{\sin u}{\csc u} = \sin^2 u \) and \( \frac{\cos u}{\sec u} = \cos^2 u \). The left side becomes \( \sin^2 u + \cos^2 u \).

Verify Identity (a) using Pythagorean Identity

By the Pythagorean identity, \( \sin^2 u + \cos^2 u = 1 \). Thus, the given expression \( \frac{\sin u}{\csc u} + \frac{\cos u}{\sec u} \) simplifies directly to 1, verifying identity (a).

Simplify Left Side of Identity (b)

Notice \( 1 - \cos^2 x = \sin^2 x \) due to the Pythagorean identity. Also, \( \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \), so \( 1 + \cot^2 x = \frac{\sin^2 x + \cos^2 x}{\sin^2 x} = \frac{1}{\sin^2 x} \).

Calculate Product for Identity (b)

The product \( (1 - \cos^2 x)(1 + \cot^2 x) = \sin^2 x \cdot \frac{1}{\sin^2 x} = 1 \), confirming identity (b).

Simplify Left Side of Identity (c)

Begin by simplifying \( \csc t = \frac{1}{\sin t} \). Therefore, \( \csc t - \sin t = \frac{1}{\sin t} - \sin t \) which becomes \( \frac{1 - \sin^2 t}{\sin t} = \frac{\cos^2 t}{\sin t} \).

Verify Identity (c)

Multiply \( \sin t \) with \( \frac{\cos^2 t}{\sin t} \) to obtain \( \cos^2 t \), confirming the left side equals the right side of identity (c).

Simplify Left Side of Identity (d)

\( \csc^2 t = 1 + \cot^2 t \). Therefore, \( 1 - \csc^2 t = -\cot^2 t \), leading to \( \frac{1-\csc^2 t}{\csc^2 t} = \frac{-\cot^2 t}{\csc^2 t} = \frac{-\cos^2 t}{\sin^2 t (1 + \cot^2 t)} \).

Verify Identity (d)

\( 1 + \cot^2 t = \frac{1}{\sin^2 t} \). Simplifying further yields \( \frac{-\cos^2 t}{1} = -\cos^2 t \) and \( \frac{-1}{\sec^2 t} = -\cos^2 t \), confirming identity (d).

Key concepts

Pythagorean Identity

The Pythagorean Trigonometric Identity is a cornerstone in trigonometry, expressing a relation among the squares of sine and cosine functions. This identity is given by:

• \( \sin^2 u + \cos^2 u = 1 \)

This fundamental relationship is derived from the Pythagorean Theorem applied to a unit circle. In the unit circle, the hypotenuse equals 1, reflecting why the identity sums to 1. This identity is particularly powerful because it allows us to express the square of one trigonometric function in terms of the other. This facilitates solving equations and verifying identities, such as in exercise (a), where simplifying \( \frac{\sin u}{\csc u} + \frac{\cos u}{\sec u} \) becomes trivial using this identity. Recognizing and employing the Pythagorean Identity can thus greatly simplify many trigonometric problems.

Trigonometric Simplification

Trigonometric simplification involves rewriting trigonometric expressions to make them more manageable and easier to work with, often using established identities. One common approach involves expressing all trigonometric functions in terms of sine and cosine, the most familiar and simplest functions.
For example, in exercise (b), we simplified \( 1-\cos^2 x \) to \( \sin^2 x \) using the Pythagorean identity. Similarly, understanding that \( \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \) simplifies the expression \( 1 + \cot^2 x \) to \( \frac{1}{\sin^2 x} \). These transformations allow us to rewrite trigonometric expressions into simpler forms, paving the way for straightforward calculation or verification.
Simplifying these expressions often involves:

• Recognizing common trigonometric identities
• Breaking down complex expressions into simpler components

Practicing these simplifications helps build skill in both computation and analysis of trigonometric functions.

Cosecant and Secant Identities

Cosecant (\( \csc \)) and secant (\( \sec \)) are reciprocal trigonometric functions related to sine (\( \sin \)) and cosine (\( \cos \)), respectively. Understanding these relationships is key in simplifying trigonometric expressions.

• \( \csc u = \frac{1}{\sin u} \)
• \( \sec u = \frac{1}{\cos u} \)

In exercise (a), these identities were crucial to simplifying \( \frac{\sin u}{\csc u} + \frac{\cos u}{\sec u} \) since the expressions simplify to \( \sin^2 u \) and \( \cos^2 u \) respectively, leading directly to using the Pythagorean Identity for proof.
These identities help convert reciprocals into products or sums of familiar functions, making them easier to manipulate and solve. Such simplifications are frequently used in verifying identities and solving complex trigonometric equations efficiently.

Cotangent Identity

The cotangent identity links the cotangent function \( \cot x \) to sine and cosine through division. It is expressed as:

• \( \cot x = \frac{\cos x}{\sin x} \)
• \( \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \)

This identity plays a significant role in simplifying expressions involving \( \cot x \), as seen in exercise (d). By understanding \( 1 + \cot^2 x = \frac{1}{\sin^2 x} \), you can transform parts of complex expressions to more manageable forms.
For complex expressions, such as \( 1 - \csc^2 t \) which simplifies to \( -\cot^2 t \), understanding cotangent's relationship can alter the approach to simplification. This also includes their utility in solving identities by enabling manipulation into alternate forms like \( -\cos^2 t \), further verifying these identities. This demonstrates the mathematical elegance and consistently reliable nature of trigonometric relationships.