Trigonometric Identitied

How does one combine fractions?

1. Multiply each side by the other side's denominator (as cos x/cos x or sin x/sin x, etc.) 2. Combine the denominator now to have a similar equation and solve for the top portion of the fraction 3. Cross out stuff if needed and do not forget to use the fundamental identities

Try to use distribution on: sec x (sec x - cos x)= tan^2 x

1. sec^2 x - (sec x ⋅ cos x)= tan^2 x ; Multiply the variables within the parentheses by the value on the outside 2. sec^2 x - (1/cos x ⋅ cos x)= tan^2 x ; Because sec x is equivalent to 1/cos x, change the variable inside so that it makes sense 3. sec^2 x - 1= tan^2 x ; 1/cos x multiplied by cos x is simply 1, because cos x is essentially cos x/1 4. sec^2 x= tan^2 x + 1 ; Because these two expressions/equations are said to be equal, you can move one (add one) to both sides 5. sec^2 x - 1= tan^2 x + 1 - 1 ; Subtract one from each side now in order to make sense of it--as sec^2 x -1 is equal to tan^2 x and 1 - 1 is essentially 0 6. tan^2 x= tan^2 x ; Add the 1 and subtract the one, and finally use your fundamental identities to finalize your solution

Try to factor: sin x cos^2 x - sin x= -sin^3 x

1. sin x (cos^2 x - 1)= -sin^3 x ; First factor out the greatest common factor on the outside and leave the rest on the inside 2. sin x (-sin^2 x)= -sin^3 x ; One of the fundamental identities is cos^2 x + sin^2 x = 1, so cos^2 x - 1 = -sin^2 x 3. -sin^3 x= -sin^3 x ; When multiplying things of the same variable (ie. sin x sin, cos x cos...), the exponents are added to one another

What is a conjugate (in math)?

A math conjugate is formed by changing the sign between two terms in a binomial. For instance, the conjugate of x + y is x - y. We can also say that x + y is a conjugate of x - y. In other words, the two binomials are conjugates of each other.

What is a monomial?

A polynomial with one term

What is an identity?

An equation/value that is true for all real values in the domain of the variable

What are the six basic ways one can verify trigonometric identities?

Distribution, Factoring, Combining Fractions, FOIL-ing, Getting a Monomial Denominator, or Eliminating a Fraction

What does FOIL stand for?

First, Outer, Inner, Last

What is the fourth thing someone should look for when verifying trigonometric identities?

If steps 1-3 haven't worked, convert all of the terms into sines and cosines (if possible)

What is the key to both verifying identities and solving equations?

It is the ability to use the fundamental identities and the rules of algebra to rewrite trigonometric expressions

What is the second thing someone should look for when verifying trigonometric identities?

Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator

What is the third thing someone should look for when verifying trigonometric identities?

Look for opportunities to use the fundamental identities (Sines and cosines pair well, secants and tangents, and cosecants and cotangents)

To try to eliminate a fraction what does one need to do?

Multiply the fraction by its conjugate

What is the first thing someone should look for when verifying trigonometric identities?

Work with one side at a time (the more complicated one first)