Question

Verify that the following are identities (see Example 6). (a) \((1+\sin z)(1-\sin z)=\frac{1}{\sec ^{2} z}\) (b) \((\sec t-1)(\sec t+1)=\tan ^{2} t\) (c) \(\sec t-\sin t \tan t=\cos t\) (d) \(\frac{\sec ^{2} t-1}{\sec ^{2} t}=\sin ^{2} t\)

Short answer

All identities are verified by simplifying both sides to equal values using trigonometric identities.

Step-by-step solution

Verify Identity (a)

The identity to verify is \((1+\sin z)(1-\sin z)=\frac{1}{\sec ^{2} z}\). First, simplify the left-hand side: \((1+\sin z)(1-\sin z)\) is a difference of squares, so it equals \(1-\sin^2 z\). Next, recall the Pythagorean identity: \(1 - \sin^2 z = \cos^2 z\). Hence, the left-hand side becomes \(\cos^2 z\). Now, the right-hand side \(\frac{1}{\sec^2 z}\) simplifies to \(\cos^2 z\) because \(\sec z = \frac{1}{\cos z}\), so \(\sec^2 z = \frac{1}{\cos^2 z}\). Therefore, both sides equal \(\cos^2 z\), and the identity is verified.

Verify Identity (b)

The identity to verify is \((\sec t-1)(\sec t+1)=\tan ^{2} t\). First, the left-hand side can be expressed as a difference of squares: \((\sec t)^2 - 1\). The Pythagorean identity \(\sec^2 t = \tan^2 t + 1\) allows us to rewrite \(\sec^2 t - 1\) as \(\tan^2 t\). Therefore, both sides of the equation are equal, confirming the identity.

Verify Identity (c)

The identity to verify is \(\sec t - \sin t \tan t = \cos t\). Start with the left-hand side. Recall the definition of secant: \(\sec t = \frac{1}{\cos t}\). Substitute it in to get \(\frac{1}{\cos t} - \sin t \tan t\). Since \(\tan t = \frac{\sin t}{\cos t}\), this could be further simplified to \(\frac{1}{\cos t} - \sin t \cdot \frac{\sin t}{\cos t}\). Combine the terms over a common denominator: \(\frac{1-\sin^2 t}{\cos t}\). Using the Pythagorean identity, \(1-\sin^2 t = \cos^2 t\). So, the expression simplifies to \(\frac{\cos^2 t}{\cos t} = \cos t\). Hence, the identity is verified.

Verify Identity (d)

The identity to verify is \(\frac{\sec ^{2} t-1}{\sec ^{2} t}=\sin ^{2} t\). Start with the left-hand side. Using the Pythagorean identity \(\sec^2 t = \tan^2 t + 1\), substitute to get \(\frac{\tan^2 t + 1 - 1}{\sec^2 t}\), which simplifies to \(\frac{\tan^2 t}{\sec^2 t}\). Recall \(\tan t = \frac{\sin t}{\cos t}\) and \(\sec t = \frac{1}{\cos t}\), thus \(\tan^2 t = \frac{\sin^2 t}{\cos^2 t}\) and \(\sec^2 t = \frac{1}{\cos^2 t}\). Therefore, \(\frac{\tan^2 t}{\sec^2 t} = \sin^2 t\). The identity is therefore verified.

Key concepts

Pythagorean Identity

Trigonometric identities like the Pythagorean identity are fundamental in solving trigonometry problems. The Pythagorean identity is an essential tool that relates different trigonometric functions to one another. This identity states that \(1 - \sin^2 z = \cos^2 z\). It shows a direct relationship between the sine and cosine of any angle. Knowing this identity can simplify many expressions and verify compound trigonometric equations. For example, when verifying the expression \( (1+\sin z)(1-\sin z) = \frac{1}{\sec^2 z} \), we apply the identity to simplify \(1 - \sin^2 z\) to \(\cos^2 z\). Similarly, in the exercise, when we assess \(\sec^2 t - 1\), we can replace it with \(\tan^2 t\) using the identity \(\sec^2 t = \tan^2 t + 1\). This identity not only streamlines calculations but also helps in the efficient verification of complex trigonometric identities.

Difference of Squares

The difference of squares is a useful algebraic expression often encountered in trigonometry. It takes the form \(a^2 - b^2 = (a + b)(a - b)\), and is instrumental in breaking down and simplifying mathematical expressions. In trigonometry, you see it applied when we encounter terms such as \(1 - \sin^2 z\) or \(\sec^2 t - 1\). These can easily be restructured as \( (1 + \sin z)(1 - \sin z) \) and \( (\sec t+1)(\sec t-1) \) respectively, illustrating the utility of the difference of squares in simplifying trigonometric expressions. This simplification allows us to utilize identities like the Pythagorean identity seamlessly, leading to straightforward verifications of identities.

Verification of Trigonometric Identities

Verifying trigonometric identities is a common task that helps confirm whether a given equation holds for all values of the variable. The process usually involves transforming one side of an equation to match the other side by using known identities, algebraic manipulations, and logical steps. For example, in verifying the identity \(\sec t - \sin t \tan t = \cos t\), we start by rewriting trigonometric functions in terms of sine and cosine: \(\sec t = \frac{1}{\cos t}\) and \(\tan t = \frac{\sin t}{\cos t}\). By substituting these into the given identity and simplifying using common denominators and known identities like the Pythagorean identity \(1 - \sin^2 t = \cos^2 t\), we can efficiently demonstrate the validity of the identity. This verification technique improves problem-solving skills and provides a deeper understanding of the relationships between trigonometric functions.