Question

What are six trig identities?

Answer 1

Hint: We should know about trigonometry at first. Trigonometry is a branch of mathematics that deals with angles of triangles. There are six identities of trigonometry. These identities are very important for solving different trigonometric problems.

Complete step by step solution:
We have the question of what are 6 trigonometric identities.
There are six trigonometric identities that are sine, cosine, tangent, secant, cosecant, and cotangent.
In mathematical terms we write these trigonometric identities as: \[\sin ,\cos ,\tan ,\sec ,\csc \] and \[\cot \].
We write these identities with an angle, they would not make any sense if we write them without angles. These trigonometric identities are related to one another. We use these relations as formulas to solve the trigonometric problems. We should know that the sine, cosine and tangent are the major identities. All the other three identities are derived from the major ones.
Let there be a triangle ABC such that:

The identities are derived from the triangle. We have to derive the six identities from the triangle. Doing so we get:
$\begin{align}
  & \sin x=\dfrac{AB}{AC} \\
 & \cos x=\dfrac{BC}{AB} \\
 & \tan x=\dfrac{AB}{BC} \\
\end{align}$
The remaining three identities are given by:
$\begin{align}
  & \csc x=\dfrac{AC}{AB} \\
 & \sec x=\dfrac{AB}{BC} \\
 & \cot x=\dfrac{BC}{AB} \\
\end{align}$
The other three identities are derived from the major one as:
\[\begin{align}
  & \sin x=\dfrac{1}{\csc x} \\
 & \cos x=\dfrac{1}{\sec x} \\
 & \tan x=\dfrac{1}{\cot x} \\
\end{align}\]
We should also know that the three trigonometric identities are also related in a very unique way. They are given as:
\[\tan x=\dfrac{\sin x}{\cos x}\]
We should also know the formulas that are used very much while solving the trigonometric problems. These formulas are:
\[\begin{align}
  & {{\sin }^{2}}x+{{\cos }^{2}}x=1 \\
 & {{\sec }^{2}}x-{{\tan }^{2}}x=1 \\
 & {{\csc }^{2}}x-{{\cot }^{2}}x=1 \\
\end{align}\]

Note: In trigonometry we should have a thorough understanding of the concepts. We should understand the concepts and have a deep understanding of identities. These identities are derived from right angled triangles. We should have learned all the formulas to solve all the questions.