How to Remember Trigonometric Identities Easily?

Quotient Identity

Let's move on to quotient identities. There are two known quotient identities we usually see in textbook references: 

• \(\tan\theta=\frac{\sin\theta}{\cos\theta}\) and 
• \(\cot\theta=\frac{\cos\theta}{\sin\theta}\)

However, any of the six functions can be a ratio between others. We can express these ratios using the tool. To find a quotient identity: 

• Start with the function we want to describe as a ratio – say tangent. 
• Then, choose two consecutive functions from it. From tangent, it would either be sine-cosine or secant-cosecant.
• Say we consider the first set. We can read the quotient identity as \(\tan\theta=\frac{\sin\theta}{\cos\theta}\). If we choose the other one, it is \(\tan\theta=\frac{\sec\theta}{\csc\theta}\).

Pythagorean Identity

Finally, we can move on to the Pythagorean identities. There are only three:

1. \(\sin^2{θ}+cos^2{θ}=1\)
2. \(tan^2{θ}+1=sec^2{θ}\)
3. \(1+cot^2{θ}=csc^2{θ}\)

To read these identities from the figure, consider the illustration again. Say we want to find the first identity \(\sin^2{θ}+cos^2{θ}=1\):

1. We consider an inverted triangle, as shown in the figure.
2. From this triangle, read the identity from top-left to top-right to bottom.
3. We read the identity as the square of the top-left function plus the square of the top-right function equals the square of the bottom function.

In this case, it is: sine squared (top-left) plus cosine squared (top-right) equals one squared (bottom): \(\sin^2{θ}+cos^2{θ}=1\)

This procedure applies to the other two inverted triangles: \(tan^2{θ}+1=sec^2{θ}\) and \(1+cot^2{θ}=csc^2{θ}\)

Summary

We can use a hexagon trigonometric tool to identify reciprocal, quotient, and Pythagorean identities (See figure)

When a line connects two directly opposite vertices, they are reciprocals. 

When we consider a function and two consecutive ones, we have a quotient identity

When we consider one of the three inverted triangles, we can derive a Pythagorean identity.