Proving Trigonometric Identities

Thread starter snowpanda
Start date Aug 8, 2010
Tags

identities Trigonometric

In summary, the purpose of proving trigonometric identities is to simplify complex equations and show the equivalence of two mathematical expressions involving trigonometric functions. To prove a trigonometric identity, one must manipulate one side of the equation using algebraic properties and trigonometric identities until it is equivalent to the other side. A trigonometric identity is considered true if both sides of the equation simplify to the same expression. Common techniques for proving trigonometric identities include using basic identities, double-angle or half-angle formulas, and symmetry and periodicity properties. Some tips for successfully proving trigonometric identities include being familiar with basic identities, using algebraic manipulations, breaking down the problem into smaller steps, and using visual aids.

Aug 8, 2010

snowpanda

Homework Statement

Prove (using the left side):

sinΘ tanΘ = cosΘ sec^2Θ - cosΘ

Homework Equations

The Attempt at a Solution

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Aug 8, 2010

HallsofIvy

Science Advisor

Homework Helper

[tex]sin(\theta)tan(\theta)= \frac{sin(\theta)}{cos(\theta)}tan(\theta)cos(\theta)= \tan^2(\theta)cos(\theta)[/tex]

Now use the fact that [itex]tan^2(\theta)= sec^2(\theta)- 1[/itex].

1. What is the purpose of proving trigonometric identities?

The purpose of proving trigonometric identities is to show that two mathematical expressions involving trigonometric functions are equivalent. This helps to simplify complex equations and can be useful in solving problems in mathematics and other fields such as physics and engineering.

2. How do you prove a trigonometric identity?

To prove a trigonometric identity, you need to manipulate one side of the equation using algebraic properties and trigonometric identities until it is equivalent to the other side. This can involve using basic trigonometric identities, such as the Pythagorean identities, as well as more advanced techniques like using double-angle or half-angle formulas.

3. How do you know if a trigonometric identity is true?

A trigonometric identity is considered true if both sides of the equation simplify to the same expression. This can be shown by using algebraic manipulations and substituting in values for the variables to verify that both sides of the equation are equal.

4. Are there any common techniques for proving trigonometric identities?

Yes, there are several common techniques for proving trigonometric identities. These include using basic trigonometric identities, using double-angle or half-angle formulas, rewriting expressions in terms of sine and cosine, and using symmetry and periodicity properties of trigonometric functions.

5. What are some tips for successfully proving trigonometric identities?

Some tips for successfully proving trigonometric identities include being familiar with basic trigonometric identities, being comfortable with algebraic manipulations, and breaking down the problem into smaller, more manageable steps. It can also be helpful to use visual aids, such as graphs or diagrams, to better understand the identities and how they relate to each other.