How do I differentiate the trigonometric functions sin(x) and cos(x) ?

As can be seen by studying the gradients of sin(x) at key points such as 0, 90, 180, 270, 360 degrees, the gradient of sin(x) at any point, x, is equal to cos(x). [This would be shown on sketches of the functions on the whiteboard.] Therefore, recalling that the derivative of a function gives the gradient of said function, we can conclude that the derivative of sin(x) is cos(x). Repeating this process for cos(x), we see that the derivative of cos(x) is -sin(x), the derivative of -sin(x) is -cos(x) and the derivative of -cos(x) is sin(x). So simply by studying the graphs of these functions we can show that differentiating sin(x) or cos(x) results in a cycle: sin(x) -> cos(x) -> -sin(x) -> -cos(x) -> back to sin(x) again. This cycle can then be memorised and the learning compounded by practise questions involving differentiating these functions. Further learning would involve differentiating tan(x), which would require the student to know the quotient rule for differentiation and the following trigonometric identities:tan(x) = sin(x) / cos(x)sin²(x) + cos²(x) = 11 / cos(x) = sec(x)