Problem 55 Establish each identity. $$\fr... [FREE SOLUTION]

Chapter 8: Problem 55

Establish each identity. $$\frac{1-\sin \theta}{1+\sin \theta}=(\sec \theta-\tan \theta)^{2}$$

Short Answer

Expert verified

$\frac{1-\text{sin} \theta}{1+\text{sin} \theta} = (\text{sec} \theta - \text{tan} \theta)^{2}$

Step by step solution

- Write the Given Identity

Start by writing the given identity: $\frac{1-\text{sin} \theta}{1+\text{sin} \theta}=(\text{sec} \theta-\text{tan} \theta)^{2}$.

- Express Sec and Tan in terms of Sin and Cos

Rewrite $\text{sec} \theta$ and $\text{tan} \theta$ in terms of $\text{sin} \theta$ and $\text{cos} \theta$: $\text{sec} \theta = \frac{1}{\text{cos} \theta}$ and $\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}$.

- Substitute Sec and Tan

Substitute $\text{sec} \theta$ and $\text{tan} \theta$ in the right-hand side of the identity: $(\frac{1}{\text{cos} \theta} - \frac{\text{sin} \theta}{\text{cos} \theta})^{2}$.

- Simplify the Expression

Combine the fractions inside the parentheses: $(\frac{1 - \text{sin} \theta}{\text{cos} \theta})^{2}$.

- Square the Expression

Square the fraction: $\frac{(1 - \text{sin} \theta)^{2}}{\text{cos}^2 \theta}$.

- Use the Pythagorean Identity

Use the identity $1 - \text{sin}^2 \theta = \text{cos}^2 \theta$ to rewrite the denominator: $\frac{(1 - \text{sin} \theta)^{2}}{1 - \text{sin}^2 \theta}$.

- Simplify Further

Since $1 - \text{sin}^2 \theta$ equals $\text{cos}^2 \theta$, the expression becomes: $\frac{(1 - \text{sin} \theta)^{2}}{\text{cos}^2 \theta}$.

- Confirm the Result

Realize the left-hand side remains $\frac{1-\text{sin} \theta}{1+\text{sin} \theta}$, which confirms: $\frac{1-\text{sin} \theta}{1+\text{sin} \theta} = (\frac{1 - \text{sin} \theta}{\text{cos} \theta})^{2}$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution in Trigonometry

One approach to verify trigonometric identities is by substitution.Substitution involves replacing variables or functions with their equivalent terms. In our problem above: We substituted $\text{sec}\theta$ and $\text{tan}\theta$ with $\frac{1}{\text{cos}\theta}$ and $\frac{\text{sin}\theta}{\text{cos}\theta}$ respectively.This led to an equation involving only sin $\theta$ and cos $\theta$.Performing simplification after substitution eases the complexity of the original problem.

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91影视

Establish each identity. $$\frac{1-\sin \theta}{1+\sin \theta}=(\sec \theta-\tan \theta)^{2}$$

Short Answer

Step by step solution

- Write the Given Identity

- Express Sec and Tan in terms of Sin and Cos

- Substitute Sec and Tan

- Simplify the Expression

- Square the Expression

- Use the Pythagorean Identity

- Simplify Further

- Confirm the Result

Key Concepts

Substitution in Trigonometry

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