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Trigonometry

Introduction

An application where we can use the techniques we learned to simplify trigonometric expressions is proving trigonometric identities. An identity is an equation that holds true regardless of the value(s) of its variable(s). While there are many identities already proven and commonly used to simplify expressions, we can be given identities to prove using algebra, formulas, basic trigonometric ratios, etc.

The general method of proving trigonometric identities is to work on each side of the equation separately, and simplify or manipulate each side until you reach the same expression on both sides.

Example

For example, suppose we are asked to prove the identity:

\(sinx\cdot secx = tanx\),

We can simplify the left side of the equation separately as follows:

\(sinx\cdot secx\)

                                                                                                       \(=sinx\cdot \frac{1}{cosx}\)            since \(secx=\frac{1}{cosx}\)

\(=\frac{sinx}{cosx}\)

                                                                                                           \(=tanx\)                  since \(\frac{sinx}{cosx}=tanx\)

We're done once we've reached the same expression on both sides of the equation, specifically \(tanx\). In this example, we didn't have to manipulate the right side of the equation in order to prove the identity.

Strategies

Here are some additional tips/strategies to use when proving trig identities:

  • Start with the more complicated side, this will give you more opportunities to apply formulas or identities and simplify the expression. Note: Sometimes, you may not need to simplify one of the sides!
  • Once you've picked a side to start working on, see if the given expression can be simplified immediately using any of the trigonometric formulas. This will make the following steps easier to work with as you will have fewer and/or simpler terms.
  • Be aware of what is and what isn't on both sides of the equation.
    • For example, if you see \(sinx\) on the left hand side but not on the right hand side, that's a sign that you might have to manipulate one of the sides so that it becomes an expression with/without \(sinx\). 
    • Another example: Suppose you see a single term on one side of the equation, but more than one on the other side, perhaps that's a sign that we need to combine like terms or substitute the expression with a single term using a formula/identity!
  • Try out various algebraic methods to manipulate the trigonometric expressions you're simplifying. We can factor using techniques like difference of squares or common factoring, or simplify by combining like terms!
  • Sometimes, simple is better! Try expressing other trigonometric functions only using sine or cosine, this could make the expression easier to simplify.

We'll apply some of these strategies in the example below!

Examples

Example: Prove the following trigonometric identities:
1. \(\frac{1-sinx}{sinxcotx}=\frac{cosx}{1+sinx}\) (starts at 0:00)
2. \(\frac{sec^2\theta}{cot\theta}-tan^3\theta=tan\theta\) (starts at 3:45)

Solution:

See this video for the solutions to the above example!

Practice

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