# An Easy Way To Learn The Basic Trigonometric Identities

First learn the structure. Learn the positions of the six trigonometry functions. First comes the $\sin$ function, underneath it comes the $\cos$ function. Then in the next column comes the $\csc$ function, underneath it comes the $\sec$ function. And in the last column comes the $\cot$ function and underneath it comes the $\tan$ function. Learning this position is important and just one time. The six trig functions are specially placed in the above given places, so as to serve our need. Observe the second diagram. There is a cross placed in between $\sin$ and $\csc$ functions. This means that $\sin$ and $\csc$ are reciprocals of each other. That means $\sin \theta = \displaystyle\frac{1}{\csc \theta}$ and $\csc \theta = \displaystyle \frac{1}{\sin \theta}$

and  hence $\csc \theta \times \sin \theta = 1$ .

Similarly there is a cross in between $\cos$ and $\sec$ functions and hence they to are reciprocals of each other and their multiplication is 1. And the same with $\tan$ and $\cot$ functions.

Thus: $\sin \theta = \displaystyle\frac{1}{\csc \theta}$ and $\csc \theta = \displaystyle \frac{1}{\sin \theta}$ and  hence $\csc \theta \times \sin \theta = 1$ . In the third diagram the functions which are connected with a line are complementary to each other and have $90 - \theta$ relationship. As we can see $\sin$ and $\cos$ are connected with a line and thus $\sin(90 - \theta) = \cos \theta$; $\cos(90 - \theta) = \sin \theta$.

In the same way if you want to find $90 - \theta$ of any function, see which function is either above it or below it. In other words all the functions in the same column are complementary functions.

As you can see in the above diagram, all the functions have a square sign, because addition formulae are always expressed in squares. But remember the position does not change. Here $\sin$ and $\cos$ functions which lie in the same column are connected with the addition formula $\sin^2 \theta + \cos^2 \theta = 1$. See how wonderfully all the three addition formulae are shown in the diagram. You can also derive formulae like $1 - \cos^2 \theta = \sin^2 \theta$ from the above formula and hence no need to learn them separately. The above diagram also gives us other two addition formula $1 + \cot^2 \theta = \csc^2 \theta$ and $1 + \tan^2 \theta = \sec^2 \theta$.

The best thing about the above diagrams is that you need to just remember the positions of the trig functions which function near them have the reciprocal relation, $90 - \theta$ functions and the addition functions. And also remember to write a square sign in case of addition function. Just for your easy understanding the interconnected functions are shown in same colour.

The Author

My name is Sohael Babwani from Mumbai, India. I give private tuitions to schoolchildren. I teach all subjects including Maths. In the year 2004, I had written an article “An Extended Approach to the Julian and the Gregorian Calendar.” This article was published in The Mathematical Gazette, London. I have made a blog/website http://www.babwani-congruence.blogspot.in/ explaining the same.