The basic trigonometric identities are formed based on our understanding of the unit circle, reference triangles and angles.

In mathematics, an “identity” is an equation which is always true, as nicely stated by trigidentities.info.

There are six trigonometric ratios, sine, cosine, tangent, cosecant, secant and cotangent.

These six trigonometric ratios are abbreviated as sin, cos, tan, csc, sec, cot.

There are fundamentally six trigonometry ratios utilized for finding the components in Trigonometry. They are called trigonometric ratios or functions. The six trigonometric ratios are sine, cosine, secant, and their reciprocals are cosecant, tangent, and cotangent, respectively. If we use the right angle triangle as a kind of perspective, then, the trigonometric ratios derived as:

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

sec θ = Hypotenuse/Adjacent Side

tan θ = Opposite Side/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

**TRIGONOMETRY RECIPROCAL IDENTITIES**:

The Reciprocal Identities of the right angle triangle are given as:

cosec θ = 1/sin θ

cot θ = 1/tan θ

sec θ = 1/cos θ

cos θ = 1/sec θ

sin θ = 1/cosec θ

tan θ = 1/cot θ

Identities and equations compared: An identity is a statement that is always true, whereas an equation is only true under certain conditions. For example

3x + 2x = 5x

is an identity that is always true, no matter what the value of x, whereas

3x = 15

is an equation (or more precisely, a conditional equation) that is only true if x = 5.

A Trigonometric identity is an identity that contains the trigonometric functions sin, cos, tan, cot, sec or csc. Trigonometric identities can be used to:

Simplify trigonometric expressions.

Solve trigonometric equations.

Prove that one trigonometric expression is equivalent to another, so that we can replace the first expression by the second expression. The second expression can give us new insights into some application that the first one doesn’t show.

Basic and Pythagorean Identities

csc(x)=1/ sin(x)

```
sin(x)=1/csc(x)
sec(x)=1/cos(x)
```

cos(x)=1/sec(x)

cot(x)=1/tan(x)= cos(x)// /sin(x)

tan(x)=1/cot(x)= sin(x)/Cos(x)

Notice how a “co-(something)” trig ratio is always the reciprocal of some “non-co” ratio. You can use this fact to help you keep straight that cosecant goes with sine and secant goes with cosine.

The following (particularly the first of the three below) are called “Pythagorean” identities.

sin2(t) + cos2(t) = 1

tan2(t) + 1 = sec2(t)

1 + cot2(t) = csc2(t)

Note that the three identities above all involve squaring and the number 1. You can see the Pythagorean-Thereom relationship clearly if you consider the unit circle, where the angle is t, the “opposite” side is sin(t) = y, the “adjacent” side is cos(t) = x, and the hypotenuse is 1.

(You may have noticed the radicals on the 1’s in the above. Yes, these simplify to just 1, so you can write things that way, too, and you certainly should do the simplification in your final hand-in answer. But notice how all the denominators are 2’s, and how the numerators go up or go down, 1, 2, 3. This can be helpful for remembering the trig values.)

You might be given a complete unit circle, with the values for the angles in the other three quadrants, too. But you only need to know the values in the first quadrant. Once you know them, and because the values repeat (other than sign) in the other quadrants, you know everything you need to know about the unit circle