An identity is an equality which remains true for entire values of the variables involved in the equation. The algebraic identity \(\mathbf{(a+b)^2 = a^2~+~2ab~+~b^2}\)

Thus, for an identity \(X\)

## Trigonometric Identities

Similarly, any equation which involves trigonometric ratios of an angle represents a trigonometric identity.

The upcoming discussion covers the fundamental trigonometric identities and their proofs.

Consider a right angle \(∆ABC\)

Applying Pythagoras Theorem in \(∆ABC\)

\((hypotenuse)^2\)

\(AC^2 = AB^2~+~BC^2 ~~~~~~~~~~~~~~~~~~~—(1)\)

Now, divide each term of equation (1) by \(AC^2\)

\(\frac{AC^2}{AC^2}\)

\(⇒\frac{AB^2}{AC^2}~+~\frac{BC^2}{AC^2}\)

\(⇒(\frac{AB}{AC})^2~+~(\frac{BC}{AC})^2\)

Making use of trigonometric ratios; for \(∆ABC\)

\(\frac{AB}{AC}\)

Similarly,

\(\frac{BC}{AC}\)

Replacing the values of \(\frac{AB}{AC}\)

\(\mathbf{sin^2~A~+~cos^2~A = 1}\)

This is valid for all the values of \(∠A\)

Now Dividing the equation (1) by \(AB^2\)

\(\frac{AC^2}{AB^2}\)

\(⇒\frac{AC^2}{AB^2}\)

By referring trigonometric ratios, it can be seen that:

\(\frac{AC}{AB}\)

Similarly,

\(\frac{BC}{AB}\)

Replacing the values of \(\frac{AC}{AB}\)

\(\mathbf{1~+~tan^2~A = sec^2~A}\)

As it is known that \(tan~A\)

Dividing the equation (1) by \(BC^2\)

\(\frac{AC^2}{BC^2}\)

\(⇒\frac{AC^2}{BC^2}\)

\(⇒(\frac{AC}{BC})^2\)

By referring trigonometric ratios, it can be seen that:

\(\frac{AC}{BC}\)

Also,

\(\frac{AB}{BC}\)

Replacing the values of \(\frac{AC}{BC}\)

\(\mathbf{cosec~^2~A = 1~+~cot^2~A}\)

Since \(cosec~A\)

Thus, the three fundamental trigonometric identities can be listed as:

- \(sin^2~A~+~cos^2~A\)
= \(1\) - \(1~+~tan^2~A\)
= \(sec^2~A\) - \(1~+~cot^2~A\)
= \(cosec^2~A\)

Example: Consider a \(∆ABC\)

\(BC\)

As the length of perpendicular and base is given; it can be concluded that,

\(tan~A\)

Now, using the trigonometric identity \(1~+~tan^2~A\)

\(sec^2~A\)

\(sec^2~A\)

\(⇒sec~A\)

Since, the ratio of lengths is positive, we can neglect \(sec~A\)

Therefore, \(sec~A\)

Using these identities, we can solve various mathematical problems. All you need to know about trigonometry and its applications is just a click away. Visit BYJU’S to learn more.

‘

**Practise This Question**