What is $\frac{{\sin \theta - \cos \theta + 1}}{{\sin \theta + \cos \theta - 1}} - \frac{{\sin \theta + 1}}{{\cos \theta }}$ equal to?

Question

Prepp Team · Accepted Answer

Simplifying the Trigonometric Expression

We are asked to simplify the expression:

$\frac{{\sin \theta - \cos \theta + 1}}{{\sin \theta + \cos \theta - 1}} - \frac{{\sin \theta + 1}}{{\cos \theta }}$

This expression involves trigonometric functions $\sin \theta$ and $\cos \theta$. To simplify this, we can first focus on the first term of the expression and try to simplify it.

Simplifying the First Term

The first term is $\frac{{\sin \theta - \cos \theta + 1}}{{\sin \theta + \cos \theta - 1}}$. Let's rearrange the terms in the numerator and the denominator to identify a pattern that allows for simplification:

Numerator: $\sin \theta - \cos \theta + 1 = (\sin \theta + 1) - \cos \theta$
Denominator: $\sin \theta + \cos \theta - 1 = (\sin \theta + \cos \theta) - 1$

We can use an algebraic technique combined with trigonometric identities. Let's multiply the numerator and the denominator of the first term by $(\sin \theta + \cos \theta) + 1$. This choice is motivated by the denominator's structure $(\sin \theta + \cos \theta) - 1$, aiming to use the difference of squares formula $(A-B)(A+B) = A^2 - B^2$.

The first term becomes:

$\frac{(\sin \theta - \cos \theta + 1)(\sin \theta + \cos \theta + 1)}{(\sin \theta + \cos \theta - 1)(\sin \theta + \cos \theta + 1)}$

Simplifying the Denominator

Let $A = \sin \theta + \cos \theta$ and $B = 1$. The denominator is $(A - B)(A + B)$. Using the identity $(A-B)(A+B) = A^2 - B^2$:

$(\sin \theta + \cos \theta - 1)(\sin \theta + \cos \theta + 1) = (\sin \theta + \cos \theta)^2 - 1^2$

Now, expand $(\sin \theta + \cos \theta)^2$ using the identity $(a+b)^2 = a^2 + 2ab + b^2$:

$(\sin \theta + \cos \theta)^2 - 1 = (\sin^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta) - 1$

Using the fundamental Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$:

$(1 + 2\sin \theta \cos \theta) - 1 = 2\sin \theta \cos \theta$

So, the denominator simplifies to $2\sin \theta \cos \theta$.

Simplifying the Numerator

The numerator is $(\sin \theta - \cos \theta + 1)(\sin \theta + \cos \theta + 1)$. We can rearrange this as $((\sin \theta + 1) - \cos \theta)((\sin \theta + 1) + \cos \theta)$. This is again in the form $(A - B)(A + B)$ where $A = \sin \theta + 1$ and $B = \cos \theta$.

Using the identity $(A-B)(A+B) = A^2 - B^2$:

$((\sin \theta + 1) - \cos \theta)((\sin \theta + 1) + \cos \theta) = (\sin \theta + 1)^2 - \cos^2 \theta$

Expand $(\sin \theta + 1)^2$ using the identity $(a+b)^2 = a^2 + 2ab + b^2$:

$(\sin^2 \theta + 2\sin \theta + 1) - \cos^2 \theta$

Rearrange terms and use the identity $1 = \sin^2 \theta + \cos^2 \theta$, which implies $1 - \cos^2 \theta = \sin^2 \theta$:

$\sin^2 \theta + 2\sin \theta + (1 - \cos^2 \theta) = \sin^2 \theta + 2\sin \theta + \sin^2 \theta$

$= 2\sin^2 \theta + 2\sin \theta$

Factor out the common term $2\sin \theta$:

$= 2\sin \theta (\sin \theta + 1)$

So, the numerator simplifies to $2\sin \theta (1 + \sin \theta)$.

Putting the Simplified First Term Together

Now, substitute the simplified numerator and denominator back into the first term of the original expression:

$\frac{2\sin \theta (1 + \sin \theta)}{2\sin \theta \cos \theta}$

Assuming $\sin \theta \neq 0$, we can cancel out the common factor $2\sin \theta$ from the numerator and the denominator:

$= \frac{1 + \sin \theta}{\cos \theta}$

Thus, the first term $\frac{{\sin \theta - \cos \theta + 1}}{{\sin \theta + \cos \theta - 1}}$ simplifies to $\frac{1 + \sin \theta}{\cos \theta}$.

Substituting Back into the Original Expression

The original expression is $\frac{{\sin \theta - \cos \theta + 1}}{{\sin \theta + \cos \theta - 1}} - \frac{{\sin \theta + 1}}{{\cos \theta }}$. Now, replace the first term with its simplified form $\frac{1 + \sin \theta}{\cos \theta}$:

$\frac{1 + \sin \theta}{\cos \theta} - \frac{\sin \theta + 1}{\cos \theta}$

Since the two fractions have the same denominator $\cos \theta$ (assuming $\cos \theta \neq 0$), we can subtract the numerators:

$\frac{(1 + \sin \theta) - (\sin \theta + 1)}{\cos \theta}$

Remove the parentheses in the numerator:

$= \frac{1 + \sin \theta - \sin \theta - 1}{\cos \theta}$

Combine the terms in the numerator:

$= \frac{(1 - 1) + (\sin \theta - \sin \theta)}{\cos \theta}$

$= \frac{0 + 0}{\cos \theta}$

$= \frac{0}{\cos \theta}$

Assuming $\cos \theta \neq 0$, any fraction with a numerator of $0$ and a non-zero denominator is equal to $0$.

$= 0$

The simplified value of the given trigonometric expression is $0$.

Revision Table: Essential Trigonometric and Algebraic Identities

Mastering trigonometric and algebraic identities is key to simplifying complex expressions. Below are some identities used in this problem:

Type	Identity	Formula
Pythagorean	Fundamental Identity	$\sin^2 \theta + \cos^2 \theta = 1$
Algebraic	Square of a Sum	$(a+b)^2 = a^2 + 2ab + b^2$
Algebraic	Difference of Squares	$(a-b)(a+b) = a^2 - b^2$

Additional Information: Strategies for Simplifying Trigonometric Expressions

When faced with a trigonometric expression to simplify, consider these common strategies:

Convert to Sine and Cosine: Rewrite all terms using only $\sin \theta$ and $\cos \theta$. This can often make common factors or identities more apparent.
Use Pythagorean Identities: Look for terms like $\sin^2 \theta + \cos^2 \theta$, $\sec^2 \theta - \tan^2 \theta$, or $\csc^2 \theta - \cot^2 \theta$, which can be replaced by constants or simpler terms.
Factor or Expand: Apply algebraic techniques like factoring (e.g., factoring out common terms, difference of squares, perfect squares) or expanding expressions.
Find a Common Denominator: If the expression involves fractions, combine them by finding a common denominator.
Multiply by a Conjugate: If the expression contains terms like $1 \pm \sin \theta$, $1 \pm \cos \theta$, or combinations involving other trig functions, multiplying the numerator and denominator by the conjugate can sometimes lead to simplification using Pythagorean identities.
Use Sum/Difference, Double Angle, Half-Angle Formulas: For more complex arguments or powers, these formulas can help transform the expression.

Solving trigonometric simplification problems requires practice and familiarity with the various identities and algebraic manipulation techniques.

Prepp Team · Answer

1

Prepp Team · Answer

2 sin &theta;

Prepp Team · Answer

2 cos &theta;

Simplifying the Trigonometric Expression

Simplifying the First Term

Simplifying the Denominator

Simplifying the Numerator

Putting the Simplified First Term Together

Substituting Back into the Original Expression

Revision Table: Essential Trigonometric and Algebraic Identities

Additional Information: Strategies for Simplifying Trigonometric Expressions

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Type	Identity	Formula
Pythagorean	Fundamental Identity	\(\sin^2 \theta + \cos^2 \theta = 1\)
Algebraic	Square of a Sum	\((a+b)^2 = a^2 + 2ab + b^2\)
Algebraic	Difference of Squares	\((a-b)(a+b) = a^2 - b^2\)