Recall the Pythegorean Identity: $$ cos^2\theta + sin^2\theta = 1^2 $$ $$ cot^2\theta + 1^2 = csc^2\theta $$ $$ tan^2\theta + 1^2 = sec^2\theta $$
There's a reason why it's called "Pythegorean Identity". It all starts with the unit circle. The interactive diagram below shows the top right half of a unit circle and visualization of all the trigonometric functions as lines in relation to this unit circle. Move the red circle to observe how each line changes.
The first thing to look at is the \(cos\) and \(sin\) lines. Using Pytheoras Theorum, we can already identify the most important identity: $$cos^2\theta + sin^2\theta = 1^2 $$ From the lines, you can also see that: $$ cot^2\theta + 1^2 = csc^2\theta $$ This is interesting because we can also derive it from the first formula: $$ cos^2\theta + sin^2\theta = 1^2 $$ $$ \frac{cos^2\theta}{sin^2\theta} + \frac{sin^2\theta}{sin^2\theta} = \frac{1^2}{sin^2\theta} $$ $$ cot^2\theta + 1^2 = csc^2\theta $$ Likewise, we can also observe from the lines that: $$ tan^2\theta + 1^2 = sec^2\theta $$ Similarly, we can also derive it from the first formula! $$ cos^2\theta + sin^2\theta = 1^2 $$ $$ \frac{cos^2\theta}{cos^2\theta} + \frac{sin^2\theta}{cos^2\theta} = \frac{1^2}{cos^2\theta} $$ $$ \frac{sin^2\theta}{cos^2\theta} + \frac{cos^2\theta}{cos^2\theta} = \frac{1^2}{cos^2\theta} $$ $$ tan^2\theta + 1^2 = sec^2\theta $$
And there you have it, all 3 identities derived!