Problem 57 - Algebra and Trigonometry

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: both sides simplify to \(2 + \sec t \csc t\).

Step 1: Expand the Left Side

First, we need to expand \((\sin t + \cos t)^2\). Expanding gives:\[(\sin t + \cos t)^2 = \sin^2 t + 2\sin t \cos t + \cos^2 t.\]

Step 2: Simplify the Expression

Using the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\), we simplify the expression from Step 1:\[\sin^2 t + 2\sin t \cos t + \cos^2 t = 1 + 2\sin t \cos t.\]

Step 3: Substitute Back into Original Equation

Substitute back into the original equation:\[\frac{1 + 2\sin t \cos t}{\sin t \cos t}.\]

Step 4: Simplify the Fraction

Separate the fraction:\[\frac{1}{\sin t \cos t} + \frac{2\sin t \cos t}{\sin t \cos t}.\]This simplifies to:\[\frac{1}{\sin t \cos t} + 2.\]

Step 5: Use Reciprocal Identities

Use the reciprocal identities \(\frac{1}{\sin t} = \csc t\) and \(\frac{1}{\cos t} = \sec t\) on the first term:\[\frac{1}{\sin t \cos t} = \sec t \csc t.\]

Step 6: Combine Results

Combine the simplified results:\[2 + \sec t \csc t.\]This shows that the left side equals the right side of the identity.

Key Concepts

Pythagorean identity

One of the fundamental concepts in trigonometry is the Pythagorean identity, which relates the squares of sine and cosine functions to each other. The identity is given by: \[ \sin^2 t + \cos^2 t = 1 \] This identity essentially states that, for any angle \(t\), the sum of the squares of its sine and cosine will always equal one. This can be visualized easily if imagined in the context of a right triangle drawn within a unit circle. Here, the lengths of the legs of the triangle are the values of \(\sin t\) and \(\cos t\), and the hypotenuse is always 1 (the radius of the unit circle).
Understanding this identity is crucial because it helps in simplifying many trigonometric expressions, as we have seen in our exercise. By substituting \(\sin^2 t + \cos^2 t\) with 1, we can often simplify more complex trigonometric equations, making them easier to solve or verify.
In our step-by-step solution, we utilized this identity to simplify \((\sin t + \cos t)^2\) by recognizing that \(\sin^2 t + \cos^2 t\) simplifies to 1, helping us progress toward verifying the given identity.

Reciprocal identities

Reciprocal identities are another pillar of trigonometry, providing a way to express trigonometric functions in terms of the reciprocal of other functions. These identities are useful for rewriting expressions and simplifying calculations. The main reciprocal identities are:

• \( \csc t = \frac{1}{\sin t} \)
• \( \sec t = \frac{1}{\cos t} \)
• \( \cot t = \frac{1}{\tan t} \)

In the context of our exercise, we applied the reciprocal identities to transform the expression \(\frac{1}{\sin t \cos t}\). By recognizing this as the product \(\sec t \csc t\), we smoothly converted a complex fraction into a familiar trigonometric form.
This transformation was key in simplifying the fractional expression, enabling an easier comparison to the right-hand side of the original identity \(2 + \sec t \csc t\). Understanding and using reciprocal identities help to bridge various trigonometric forms and are a vital tool for solving trigonometric equations.

Trigonometric expressions

Working with trigonometric expressions often involves manipulating complex forms into simpler or more recognizable patterns. A trigonometric expression can include functions like sine, cosine, tangent, cotangent, secant, and cosecant in various combinations and powers.
There are various strategies to simplify these expressions:

• Using identities such as the Pythagorean identity or reciprocal identities to transform expressions.
• Expanding products and simplifying terms to combine like terms or reduce fractions.
• Rearranging terms to match a desired form, as was necessary in our given identity exercise.

For instance, in our problem, we started by expanding \((\sin t + \cos t)^2\), which is a common technique to simplify expressions. Through expansion and using known identities, the expression was broken down and reconstructed to verify that both sides of the identity were indeed equal.
Mastering the manipulation of trigonometric expressions is foundational for anyone studying trigonometry and assists in solving complex problems more easily. With practice, recognizing when and which identities to apply becomes intuitive, making trigonometry a less daunting subject.