Problem 72 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\cos \theta}{1-\sin \theta}=\sec \theta+\tan \theta $$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \(\frac{1 + \sin \theta}{\cos \theta}\).

Step 1: Rewrite in terms of sine and cosine

To verify the identity, express all trigonometric functions in terms of sine and cosine. The right side is \(\sec \theta + \tan \theta\), which can be written as \(\frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta}\). This becomes \(\frac{1 + \sin \theta}{\cos \theta}\).

Step 2: Combine the terms on the right

Combine the fractions on the right-hand side. Since they have the same denominator, add the numerators: \(\frac{1 + \sin \theta}{\cos \theta}\).

Step 3: Simplify the identity

Rewrite the left side of the equation, \(\frac{\cos \theta}{1 - \sin \theta}\), and check whether it can be transformed into the expression of the right side. Multiply the numerator and denominator by the conjugate of the denominator: \(\frac{\cos \theta (1 + \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)}\).

Step 4: Simplify the denominator using difference of squares

Calculate the denominator: \((1 - \sin \theta)(1 + \sin \theta)\) equals \(1 - \sin^2 \theta\), which simplifies to \(\cos^2 \theta\).

Step 5: Simplify the entire expression

The fraction now looks like \(\frac{\cos \theta (1 + \sin \theta)}{\cos^2 \theta}\). Split this into two fractions: \(\frac{\cos \theta}{\cos^2 \theta}(1 + \sin \theta)\) which simplifies to \(\frac{1}{\cos \theta}(1 + \sin \theta)\).

Step 6: Show equivalence to the target expression

The expression \(\frac{1}{\cos \theta}(1 + \sin \theta)\) simplifies to \(\frac{1 + \sin \theta}{\cos \theta}\). This matches the form calculated for the right side, \(\sec \theta + \tan \theta = \frac{1 + \sin \theta}{\cos \theta}\).

Step 7: Conclude verification

Since both sides simplify to the same expression, the identity is verified.

Key Concepts

Sine and Cosine

The sine and cosine functions are fundamental in trigonometry. They are both defined based on the unit circle. The sine of an angle, \(\sin \theta\), represents the y-coordinate of a point on the unit circle, while cosine, \(\cos \theta\), represents the x-coordinate. These two functions are often considered building blocks for other trigonometric functions.

To change any trigonometric expression into its simplest form, it can be rewritten using sine and cosine. For instance, in the identity exercise, \(\frac{\cos \theta}{1-\sin \theta}\), both the given left-hand side and the right-hand side, \(\sec \theta + \tan \theta\), are expressed in terms of sine and cosine to help verify the identity.

Secant and Tangent

Secant and tangent are derived from sine and cosine. Secant, \(\sec \theta\), is the reciprocal of cosine, i.e., \(\sec \theta = \frac{1}{\cos \theta}\). Tangent, \(\tan \theta\), is the ratio of sine to cosine, \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). These functions are helpful in several trigonometric transformations.

In the problem, \(\sec \theta + \tan \theta\) is transformed from the right side to \(\frac{1+\sin \theta}{\cos \theta}\) using their definitions. This illustrates the importance of connecting complex trigonometric identities back to these basic functions.

Difference of Squares

The difference of squares is a mathematical identity that is often used to simplify algebraic expressions. It states that \(a^2 - b^2 = (a-b)(a+b)\). In trigonometry, a similar form appears with trigonometric functions.

For example, in verifying \(\frac{\cos \theta}{1-\sin \theta} = \sec \theta + \tan \theta\), we encounter \(1 - \sin^2 \theta\), which equals \(\cos^2 \theta\). This reduces the denominator by factoring it as \( (1-\sin \theta)(1+\sin \theta)\), demonstrating how trigonometric expressions can often simplify using algebraic identities.

Trigonometric Functions

Trigonometric functions include sine, cosine, tangent, secant, cosecant, and cotangent. They help model periodic phenomena and are useful in various disciplines like physics, engineering, and geometry.

Understanding each function's relationship is crucial. For instance, knowing that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) allows us to derive complex expressions from simple ones. In verifying identities, recognizing these relationships is key to transforming and proving equalities among different trigonometric forms.

Verify Identity

Verification in mathematics is the process of confirming that two sides of an equation are equivalent. When dealing with trigonometric identities, the goal is often to manipulate one or both sides of the equation into the same form.

In verifying \(\frac{\cos \theta}{1-\sin \theta} = \sec \theta + \tan \theta\), each step carefully manipulates the expressions using known identities and algebraic techniques like the difference of squares and expressions of basic trigonometric functions. This methodical approach helps in understanding and proving the identity's validity.