Problem 59 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{(\sin t+\cos t)^{2}}{\sin t \cos t}=2+\sec t \csc t$$

Step-by-Step Solution

Answer

The identity is verified as correct: \(\frac{(\sin t+\cos t)^{2}}{\sin t \cos t}=2+\sec t \csc t\).

Step 1: Expand the Numerator

The identity given is \( \frac{(\sin t + \cos t)^2}{\sin t \cos t} = 2 + \sec t \csc t \). Start by expanding the numerator \((\sin t + \cos t)^2\). This can be expanded using the formula \((a+b)^2 = a^2 + 2ab + b^2\). So we have:\[\sin^2 t + 2 \sin t \cos t + \cos^2 t\].

Step 2: Utilize Basic Trigonometric Identity

Recall the Pythagorean identity: \( \sin^2 t + \cos^2 t = 1 \). Substitute this into the expanded numerator:\[1 + 2 \sin t \cos t\].

Step 3: Simplify the Fraction

Now we re-write the fraction using the simplified numerator:\[\frac{1 + 2 \sin t \cos t}{\sin t \cos t}\].This can be split into two separate fractions:\[\frac{1}{\sin t \cos t} + \frac{2 \sin t \cos t}{\sin t \cos t}\]. Simplify this to:\[\frac{1}{\sin t \cos t} + 2\].

Step 4: Express as Separate Trigonometric Terms

The first term \( \frac{1}{\sin t \cos t} \) can be expressed as the product of two reciprocal trigonometric functions: \( \sec t = \frac{1}{\cos t} \) and \( \csc t = \frac{1}{\sin t} \). Thus,\[\frac{1}{\sin t \cos t} = \sec t \csc t\].

Step 5: Combine and Verify the Identity

After expressing the terms, the fraction becomes:\[\sec t \csc t + 2\],which is the right side of the original equation. Therefore, the identity is verified as:\[\frac{(\sin t + \cos t)^2}{\sin t \cos t} = 2 + \sec t \csc t\].

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the cornerstones of trigonometry. It states that for any angle \( t \), the sum of the square of the sine and the square of the cosine equals 1: \( \sin^2 t + \cos^2 t = 1 \).
This identity arises from the Pythagorean theorem in a right triangle, relating the sides of the triangle to the unit circle.
It's a powerful tool used to simplify complex trigonometric expressions by allowing us to replace one trigonometric function with another when terms are being squared. During simplifications, identifying and substituting \( \sin^2 t + \cos^2 t \) for 1 usually helps make the expression less complicated.
In the given exercise, using the Pythagorean identity meant that \( \sin^2 t + \cos^2 t \) was directly replaced with 1, reducing the expression and allowing further simplification.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are the inverses of the basic sine, cosine, and tangent functions.
They play a critical role in expressing more complex trigonometric functions, particularly in simplifying fractions and verifying identities.

• Secant function (\( \sec t \)) is the reciprocal of cosine: \( \sec t = \frac{1}{\cos t} \).
• Cosecant function (\( \csc t \)) is the reciprocal of sine: \( \csc t = \frac{1}{\sin t} \).

By using these reciprocal identities, expressions involving division by sine or cosine can be transformed into simpler products.
In our original problem, the term \( \frac{1}{\sin t \cos t} \) was expressed as \( \sec t \csc t \), revealing the connection between these reciprocal functions and helping to verify the identity.

Trigonometric Simplification

Trigonometric simplification is the process of altering an expression into its simplest form.
This often involves using identities to combine or rewrite terms to make expressions easier to read or solve. In trigonometry, simplification is crucial, as it often makes otherwise complex problems manageable.
Strategies for simplification include:

• Using known identities to replace or combine terms, like the Pythagorean identity.
• Breaking down terms into simpler parts, for example, splitting fractions: \( \frac{a+b}{c} = \frac{a}{c} + \frac{b}{c} \).
• Introduction of reciprocal functions to replace divisions.

In the given exercise, simplification was achieved by:

• Expanding \( (\sin t + \cos t)^2 \) and using the Pythagorean identity.
• Separating the resulting fraction into two simpler fractions, leading to easier evaluation.
• Substituting reciprocal identities to match terms with the given identity.

By simplifying expressions in these ways, the complexity is reduced, and it becomes easier to verify or solve mathematical identities.