Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ (\sec t+\tan t)^{2}=\frac{1+\sin t}{1-\sin t} $$

Step-by-Step Solution

Answer

The identity is verified.

Step 1: Expand the Left Side

Start by expanding the left side of the equation, \((\sec t + \tan t)^2 = \sec^2 t + 2\sec t \tan t + \tan^2 t\).This uses the identity \((a+b)^2 = a^2 + 2ab + b^2\).

Step 2: Use Trigonometric Identities

Recall the Pythagorean identities:\(\sec^2 t = 1 + \tan^2 t\) and \((\tan^2 t = \sec^2 t - 1)\).Substitute these into the expanded expression:\(\sec^2 t + 2\sec t \tan t + \tan^2 t = (1 + \tan^2 t) + 2\sec t \tan t + (\sec^2 t - 1)\).

Step 3: Simplify the Expression

Simplify the expression:\((1 + \tan^2 t + 2\sec t \tan t + \sec^2 t - 1) = 2\sec^2 t + 2\sec t \tan t\).Notice the term \(-1 + 1\) cancels out, simplifying the expression further.

Step 4: Factor the Right Side

Consider the right side of the identity:\(\frac{1+\sin t}{1-\sin t}\).This is equivalent to the expression one gets by multiplying the numerator and denominator by the conjugate of the denominator:\(\frac{(1+\sin t)(1+\sin t)}{(1-\sin t)(1+\sin t)}\).This results in:\(\frac{1 + 2\sin t + \sin^2 t}{1 - \sin^2 t}\).

Step 5: Simplify the Denominator using Pythagorean Identity

Simplify the denominator using the identity:\(1 - \sin^2 t = \cos^2 t\).This gives:\(\frac{1 + 2\sin t + \sin^2 t}{\cos^2 t}\).

Step 6: Recognize Sec and Tan

Split the expression up:\(\frac{1}{\cos^2 t} + \frac{2\sin t}{\cos^2 t} + \frac{\sin^2 t}{\cos^2 t}\).Notice that:\(\frac{1}{\cos^2 t} = \sec^2 t\), \(\frac{\sin t}{\cos t} = \tan t\), and \(\frac{\sin^2 t}{\cos^2 t} = \tan^2 t\).This means:\(\sec^2 t + 2\sec t \tan t + \tan^2 t\), confirming the identity.

Key Concepts

Pythagorean identities

Pythagorean identities are a set of trigonometric identities fundamental in understanding the relationships between sine, cosine, and tangent functions in trigonometry. The most well-known Pythagorean identity is:

• \( \sin^2 t + \cos^2 t = 1 \)

This identity expresses the idea that within any right-angled triangle, the square of the sine and the square of the cosine of an angle add up to one. This is analogous to the Pythagorean theorem in geometry, hence the name.
To derive other identities, we can divide each term by \( \cos^2 t \), leading us to:

• \( \tan^2 t + 1 = \sec^2 t \)

This shows the connection between the tangent and secant functions with the original sine and cosine terms.
By recognizing these identities, you can better simplify and transform equations during trigonometric proofs, such as the provided exercise.

secant function

The secant function, denoted as \( \sec t \), is a trigonometric function that is the reciprocal of the cosine function. Mathematically, it is defined as:

• \( \sec t = \frac{1}{\cos t} \)

This means that the secant of an angle is the ratio of the hypotenuse of a right triangle to the adjacent side.
In the context of identities, the secant function's square is particularly significant as it combines with the tangent function through the Pythagorean identity \( \sec^2 t = 1 + \tan^2 t \). This identity is very useful because it allows transformations between secant and tangent expressions, which is vital in simplifying or proving trigonometric equations.
The secant function itself is helpful in modeling periodic phenomena and analyzing characteristics of waves and oscillations, much like other trigonometric functions.

tangent function

The tangent function, represented as \( \tan t \), is one of the primary trigonometric functions used to express the ratio between the opposite and adjacent sides of a right triangle. It is defined by the relationship:

• \( \tan t = \frac{\sin t}{\cos t} \)

This definition reveals that the tangent is a measure of the steepness or gradient of an angle, a property that is particularly useful in slope and angle calculations.
In conjunction with the secant function, the tangent function contributes significantly to trigonometric identities, such as:

• \( \tan^2 t + 1 = \sec^2 t \)

This identity simplifies terms involving the tangent and secant functions, making it easier to navigate and solve complex trigonometric equations.
Understanding tangent's relationship with sine and cosine deepens comprehension of waves, circular motion, and various mathematical phenomena where angles and ratios are central considerations.