Problem 60 - Algebra 2

Question

Simplify each trigonometric expression. $$ (1-\sin \theta)(1+\sin \theta) \csc ^{2} \theta+1 $$

Step-by-Step Solution

Answer

The trigonometric expression simplifies to \( \csc^2\theta \)

Step 1: Apply the rule of 'the difference of squares'

To start with, apply the difference of two squares in \( (1-\sin \theta)(1+\sin \theta) \), which simplifies to \( 1 - \sin^2\theta \). Now, the expression becomes \( (1 - \sin^2\theta) \csc ^{2} \theta+1 \)

Step 2: Substitute the reciprocal identity

Next, substitute the reciprocal identity \( \csc^2\theta \) with \( \frac{1}{\sin^2\theta} \). The expression is now \( \frac{1 - \sin^2\theta}{\sin^2\theta}+1 \)

Step 3: Apply the pythagorean identity

Now, use the pythagorean identity \( 1 - \sin^2\theta = \cos^2\theta \). This gives us \( \frac{\cos^2\theta}{\sin^2\theta}+1 \), which simplifies to \( \cot^2\theta+1 \)

Step 4: Apply the pythagorean identity again

Finally, apply the pythagorean identity \( \cot^2\theta+1 = \csc^2\theta \). This simplifies the whole expression to \( \csc^2\theta \)

Key Concepts

Difference of Squares

The difference of squares is a handy algebraic identity. It lets you simplify products of two binomials like \((a-b)(a+b)\) quickly. When applied, it transforms into \(a^2 - b^2\).

In the original exercise, this identity helped simplify \((1 - \sin \theta)(1 + \sin \theta)\) to \(1 - \sin^2 \theta\). This simplification is essential for continuing to break down the trigonometric expression further. Without this step, the process would be more complicated.

Reciprocal Identities

Reciprocal identities are part of the basic trigonometric identities. They relate the main trigonometric functions to one another. The reciprocal identity for \(\csc \theta\) is \(1/\sin \theta\).

In our solution, substituting the identity \(\csc^2\theta = 1/\sin^2\theta\) was a critical move. It transformed the expression to \(\frac{1-\sin^2\theta}{\sin^2\theta} + 1\). This helped in using further identities to simplify the expression down to its simplest form.

Pythagorean Identities

Pythagorean identities are fundamental in trigonometry, offering connections between squared trigonometric functions. The most common ones are \(\sin^2\theta + \cos^2\theta = 1\), \(1 + \cot^2\theta = \csc^2\theta\), and \(1 + \tan^2\theta = \sec^2\theta\).

In the solution, the identity \(1 - \sin^2\theta = \cos^2\theta\) was used to further simplify the expression, transforming it into \(\frac{\cos^2\theta}{\sin^2\theta} + 1\). Another application was when \(\cot^2\theta + 1 = \csc^2\theta\), which helped reach the final simplified form.

Simplification of Trigonometric Expressions

Simplification is key in mathematics to express problems in their simplest form, making calculations easier.

In trigonometric expressions, combining identities like the difference of squares, reciprocal, and Pythagorean identities, allows for breaking down complex expressions. Step by step, these transformations convert an initially complicated problem into something easier to handle.

For the problem \((1-\sin \theta)(1+\sin \theta) \csc^2 \theta + 1\), the simplification journey involved multiple steps, ultimately reducing it to \(\csc^2\theta\). This approach shows the power of unraveling trigonometric expressions into manageable pieces.