Use basic identities to simplify the expression

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Trigonometric Graphs
Lessons On Trigonometry
Trigonometric Functions

The following are some common trigonometric identities: Reciprocal Identities, Quotient Identities and Pythagorean Identities. Scroll down the page for examples and solutions using the identities to simply trigonometric expressions.

Example:
Simplify \(\frac{{\sin \theta \sec \theta }}{{{{\cos }^2}\theta }}\)

Solution:
\(\frac{{\sin \theta \sec \theta }}{{{{\cos }^2}\theta }} = \frac{{\sin \theta \sec \theta }}{{\cos \theta \cos \theta }}\)
\(= \tan \theta \cdot \frac{{\sec \theta }}{{\cos \theta }}\)
= tan θ • sec2 θ
= tan θ (tan2 θ + 1)
= tan3 θ + tan θ

Example:
Simplify \(\frac{{{{\sin }^2}\theta }}{{\cos \theta }} + \frac{{{{\cos }^2}\theta }}{{\cos \theta }}\)

Solution:
\(\frac{{{{\sin }^2}\theta }}{{\cos \theta }} + \frac{{{{\cos }^2}\theta }}{{\cos \theta }}\)
\(= \frac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{\cos \theta }}\)
\(= \frac{1}{{\cos \theta }}\)
= sec θ

Simplifying Trigonometric Expressions Using Identities

Example:
(tan3x)(csc3x)

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How to Simplify Trigonometric Expressions Using Identities?

Example:
sec x cos x − cos2 x
(csc2 x − 1)(sec2 x sin2 x)

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Using Identities to Simplify Trigonometric Expressions

Example:
(csc2 x − 1)/csc2 x
(csc2 x − cot2 x)/(tan2 x - sec2 x)

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Algebraic Manipulation of Trigonometric Functions

Distributive Property, FOIL, Factoring.

Example:
cos y(tan y - sec y)
(sin x + cos x)(sin x - cos x)
sin2x cos2x + cos4x

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Algebraic Manipulation of Trigonometric Functions with fractions

Simplifying Complex Fractions, Multiplying, Dividing, Adding and Subtracting Fractions.

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Algebraic Manipulation of Trigonometric Functions - Radical Expressions

Multiplying, Dividing, Simplifying. Rationalizing the Denominator.

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Algebraic Manipulation of Trigonometric Functions - Complex Examples

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Video transcript

Let's do some examples simplifying trigonometric expressions. So let's say that I have 1 minus sine squared theta, and this whole thing times cosine squared theta. So how could I simplify this? Well the one thing that we do know-- and this is the most fundamental trig identity, this comes straight out of the unit circle-- is that cosine squared theta plus sine squared theta is equal to 1. And then, if we subtract sine squared theta from both sides, we get cosine squared theta is equal to 1 minus sine squared theta. So we have two options. We could either replace this 1 minus sine squared theta with the cosine squared theta, or we could replace this cosine squared theta with the 1 minus sine squared theta. Well I'd prefer to do the former because this is a more complicated expression. So if I can replace this with the cosine squared theta, then I think I'm simplifying this. So let's see. This will be cosine squared theta times another cosine squared theta. And so all of this is going to simplify to cosine theta times cosine theta times cosine theta times cosine theta, well, that's just going to be cosine to the fourth of theta. Let's do another example. Let's say that we have sine squared theta, all of that over 1 minus sine squared theta. What is this going to be equal to? Well we already know that 1 minus sine squared theta is the same thing as cosine squared theta. So it's going to be sine squared theta over-- this thing is the same thing as cosine squared theta, we just saw that-- over cosine squared theta, which is going to be equal to-- you could view this as sine theta over cosine theta whole quantity squared. Well what's sine over cosine? That's tangent. So this is equal to tangent squared theta. Let's do one more example. Let's say that we have cosine squared theta plus 1 minus-- actually, let's make it this way-- plus 1 plus sine squared theta. What is this going to be? Well you might be tempted, especially with the way I wrote the colors, to think, hey, is there some identity for 1 plus sine squared theta? But this is really all about rearranging it to realize that, gee, by the unit circle definition, I know what cosine squared theta plus sine squared theta is. Cosine squared theta plus sine squared theta, for any given theta, is going to be equal to 1. So this is going to be equal to 1 plus this 1 right over here, which is equal to 2.

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