We are asked to explain how we can verify a trigonometric identity. Let's start by reviewing the definition of an identity.

Identity
An equation that is true for all values of the variable is an identity and has infinitely many solutions.

A trigonometric identity is an identity involving trigonometric functions. In order to verify a trigonometric identity, we follow the same steps as if we were verifying an identity. Consider the following example.

sin^{2} θ + cos^{2} θ = 1

Let's take a look at the following diagram.

The trigonometric functions of

θ

can be substituted by their corresponding ratio.

sin θ = \frac{b}{c} cos θ = \frac{a}{c}

We can substitute these ratios into the equation and simplify to verify if it is an identity. Let's not forget that

sin^{2} θ

is equivalent to

(sin θ)^{2} .

The same applies for

cos^{2} θ .

sin^{2} θ + cos^{2} θ = ? 1

$sin θ = \frac{b}{c}$

(\frac{b}{c})^{2} + cos^{2} θ = ? 1

$cos θ = \frac{a}{c}$

(\frac{b b b}{c})^{2} + (\frac{b a b}{c})^{2} = ? 1

PowQuot

$(\frac{a}{b})^{m} = \frac{a ^{m}}{b ^{m}}$

\frac{b ^{2}}{c ^{2}} + \frac{a ^{2}}{c ^{2}} = ? 1

AddFrac

Add fractions

\frac{a ^{2} + b ^{2}}{c ^{2}} = ? 1

Since

a

and

b

are the lengths of the legs of the right triangle, we can use the Pythagorean Theorem to simplify the left-hand side of the equation.

\frac{a ^{2} + b ^{2}}{c ^{2}} = ? 1

$a^{2} + b^{2} = c^{2}$

\frac{c ^{2}}{c ^{2}} = ? 1

QuotOne

$\frac{a}{a} = 1$

1 = 1 ✓

Since we reached a true statement, the given equation is an identity.

Exercise