Editing Equation (section)

===Identities===

{{main|Identity (mathematics)|List of trigonometric identities}}

An identity is an equation that is true for all possible values of the variable(s) it contains. Many identities are known in algebra and calculus. In the process of solving an equation, an identity is often used to simplify an equation, making it more easily solvable.

In algebra, an example of an identity is the [[difference of two squares]]:

:<math>x^2 - y^2 = (x+y)(x-y) </math>

which is true for all ''x'' and ''y''.

[[Trigonometry]] is an area where many identities exist; these are useful in manipulating or solving [[trigonometric equation]]s. Two of many that involve the [[sine function|sine]] and [[cosine function|cosine]] functions are:

:<math>\sin^2(\theta)+\cos^2(\theta) = 1 </math>

and

:<math>\sin(2\theta)=2\sin(\theta) \cos(\theta) </math>

which are both true for all values of ''θ''.

For example, to solve for the value of ''θ'' that satisfies the equation:

:<math>3\sin(\theta) \cos(\theta)= 1\,, </math>

where ''θ'' is limited to between 0 and 45 degrees, one may use the above identity for the product to give:

:<math>\frac{3}{2}\sin(2 \theta) = 1\,,</math>

yielding the following solution for ''θ:''

:<math>\theta = \frac{1}{2} \arcsin\left(\frac{2}{3}\right) \approx 20.9^\circ.</math>

Since the sine function is a [[periodic function]], there are infinitely many solutions if there are no restrictions on ''θ''. In this example, restricting ''θ'' to be between 0 and 45 degrees would restrict the solution to only one number.