Differentiation of trigonometric functions
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin′("a") = cos("a"), meaning that the rate of change of sin("x") at a particular angle "x = a" is given by the cosine of that angle. All derivatives of circular trigonometric functions can be found from those of sin("x") and cos("x") by means of the quotient rule applied to functions such as tan("x") = sin("x")/cos("x"). Knowing these derivatives, the  derivatives of the inverse trigonometric functions are found using implicit differentiation. Proofs of derivatives of trigonometric functions. Limit of sin(θ)/θ as θ tends to 0. The diagram at right shows a circle with centre "O" and radius "r =" 1. Let two radii "OA" and "OB" make an arc of θ radians. Since we are considering the limit as "θ" tends to zero, we may assume "θ" is a small positive number, say 0 < θ < ½ π in the first quadrant. In the diagram, let "R" be the triangle "OAB", "R" the circular sector "OAB", and "R" the  triangle "OAC". The area of triangle "OAB" is: The area of the circular sector "OAB" is formula_2, while the area of the triangle "OAC" is given by Since each region is contained in the next, one has: Moreover, since in the first quadrant, we may divide through by ½ , giving: In the last step we took the reciprocals of the three positive terms, reversing the inequities. We conclude that for 0 < θ < ½ π, the quantity is "always" less than 1 and "always" greater than cos(θ). Thus, as "θ" gets closer to 0, is "squeezed" between a ceiling  at height 1 and a floor at height , which rises towards 1; hence sin("θ")/"θ" must tend to 1 as "θ" tends to 0 from the positive side:formula_6For the case where "θ" is a small negative number –½ π < θ < 0, we use the fact that sine is an odd function: Limit of (cos(θ)-1)/θ as θ tends to 0. The last section enables us to calculate this new limit relatively easily. This is done by employing a simple trick. In this calculation, the sign of "θ" is unimportant. Using the fact that the limit of a product is  the product of limits, and the limit result from the previous section, we find that: Limit of tan(θ)/θ as θ tends to 0. Using the limit for the sine function, the fact that the tangent function is odd, and the fact that the limit of a product is the product of limits, we find: Derivative of the sine function. We calculate the derivative of the sine function from the limit definition: Using the angle addition formula , we have: Using the limits for the sine and cosine functions: Derivative of the cosine function. From the definition of derivative. We again calculate  the derivative of the cosine function from the limit definition: Using the angle addition formula , we have: Using the limits for the sine and cosine functions: From the chain rule. To compute the derivative of the cosine function from the chain rule, first observe the following three facts: The first and the second are trigonometric identities, and the third is proven above. Using these three facts, we can write the following, We can differentiate this using the chain rule. Letting formula_21, we have: Therefore, we have proven that Derivative of the tangent function. From the definition of derivative. To calculate  the derivative of the tangent function tan "θ", we use first principles. By definition: Using the well-known angle formula , we have: Using the fact that the limit of a product is the product of the limits: Using the limit for the tangent function, and the fact that tan "δ" tends to 0 as δ tends to 0: We see immediately that: From the quotient rule. One can also compute the derivative of the tangent function using the quotient rule. The numerator can be simplified to 1 by the Pythagorean identity, giving us, Therefore, Proofs of derivatives of inverse trigonometric  functions. The following derivatives are found by setting a variable "y" equal to the inverse trigonometric function that we wish to take the derivative of. Using implicit differentiation and then solving for "dy"/"dx", the derivative of the inverse function is found in terms of "y". To convert "dy"/"dx" back into being in terms of "x", we can draw a reference triangle on the unit circle, letting "θ" be y. Using the Pythagorean theorem and the definition of the regular trigonometric functions, we can finally express "dy"/"dx" in terms of "x". Differentiating the inverse sine function. We let Where Then Taking  the derivative with respect to formula_35 on both sides and solving for dy/dx: Substituting formula_38 in from above, Substituting formula_40 in from above, Differentiating the inverse cosine function. We let Where Then Taking the derivative with respect to formula_35 on both sides and solving for dy/dx: Substituting formula_49 in from above, we get Substituting formula_51 in from above, we get Alternatively, once the derivative of formula_54 is established, the derivative of formula_55 follows immediately by differentiating the identity formula_56 so that formula_57. Differentiating the inverse tangent function. We let Where Then Taking the derivative with respect to formula_35  on both sides and solving for dy/dx: Left side: Right side: Therefore, Substituting formula_66 in from above, we get Differentiating the inverse cotangent function. We let where formula_70. Then Taking the derivative with respect to formula_35 on both sides and solving for dy/dx: Left side: Right side: Therefore, Substituting formula_77, Differentiating the inverse secant function. Using implicit differentiation. Let Then Using the chain rule. Alternatively, the derivative of arcsecant may be derived from the derivative of arccosine using the chain rule. Let Where Then, applying the chain rule to formula_88: Differentiating the inverse cosecant function. Using implicit differentiation . Let Then Using the chain rule. Alternatively, the derivative of arccosecant may be derived from the derivative of arcsine using the chain rule. Let Where Then, applying the chain rule to formula_98: