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Trigonometric identities are equations that involve trigonometric functions and are true for all values of the occurring variables for which both sides of the equality are defined. Trigonometric identities are useful for simplifying trigonometric expressions and solving trigonometric equations.

There are many different trigonometric identities, but some of the most common ones include:

* **Pythagorean identities:**

```

sin²θ + cos²θ = 1

tan²θ + 1 = sec²θ

cot²θ + 1 = cosec²θ

```

* **Angle addition and subtraction identities:**

```

sin(α + β) = sinαcosβ + cosαsinβ

sin(α - β) = sinαcosβ - cosαsinβ

cos(α + β) = cosαcosβ - sinαsinβ

cos(α - β) = cosαcosβ + sinαsinβ

tan(α + β) = (tanα + tanβ)/(1 - tanαtanβ)

tan(α - β) = (tanα - tanβ)/(1 + tanαtanβ)

```

* **Double angle and half angle identities:**

```

sin(2θ) = 2sinθcosθ

cos(2θ) = cos²θ - sin²θ

tan(2θ) = 2tanθ/(1 - tan²θ)

sin(θ/2) = √[(1 - cosθ)/2]

cos(θ/2) = √[(1 + cosθ)/2]

tan(θ/2) = √[(1 - cosθ)/(1 + cosθ)]

```

Trigonometric identities can be derived using a variety of methods, including geometric proofs, algebraic manipulations, and series expansions. They can also be found in trigonometric tables and reference books.

Here are some examples of how to use trigonometric identities to simplify trigonometric expressions and solve trigonometric equations:

**Example 1:**

Simplify the expression:

```

sin²θ + cos²θ

```

**Solution:**

Using the Pythagorean identity, we can simplify the expression as follows:

```

sin²θ + cos²θ = 1

```

**Example 2:**

Solve the equation for θ:

```

sinθcosθ = 1/2

```

**Solution:**

Dividing both sides of the equation by sinθcosθ, we get:

```

1 = 1/(2sinθcosθ)

```

Multiplying both sides of the equation by 2sinθcosθ, we get:

```

2sinθcosθ = 1

```

Using the double angle identity for sin(2θ), we can rewrite the equation as follows:

```

sin(2θ) = 1

```

The solution to the equation is θ = π/4 + nπ, where n is any integer.

Trigonometric identities are a powerful tool for working with trigonometric expressions and equations. By understanding and using trigonometric identities, we can simplify complex expressions, solve difficult equations, and gain a deeper understanding of trigonometry as a whole.

Specified formulas for trigonometry and all its operations holding true for all values is called a trigonometry identity.

Some common identities are:

(1) Pythagorean identity (sin²x × cos²x = 1)

(2) Reciprocal identity (cosec x = 1 / sin x)(cot z = 1 / tan z) (sec y = 1 / cos y)