﻿ Math formulas derivation and practice
Cotangent of the sum of two angles

The cotangent of the sum of two angles equals the product of the cotangents of the angles minus 1 divided by the sum of the cotangents of the two angles

Triple angle identity for cotangent

Triple angle identity for cotangent function is used to represent cotangent of a triple angle in terms of cotangent of one-third of the triple angle. More specifically, cotangent of a triple angle equals to the difference between three cotangent of one-third of the triple angle and cotangent of one-third of the triple angle to power three divided by 1 minus three cotangent of one-third of the triple angle to power two.

Triple angle identity for tangent

Triple angle identity for tangent function is used to represent tangent of a triple angle in terms of tangent of one-third of the triple angle. More specifically, tangent of a triple angle equals to the difference between three tangent of one-third of the triple angle and tangent of one-third of the triple angle to power three divided by 1 minus three tangent of one-third of the triple angle to power two.

Triple angle identity for cosine function

Triple angle identity for cosine function is used to represent cosine of a triple angle in terms of cosine of one-third of the triple angle. More specifically, cosine of a triple angle equals to the difference between 4 multiples of cosine of one-third of the triple angle to power three and three multiples of cosine of one-third of the triple angle.

Triple angle identity for cosine function is often used to simplify trigonometric expressions or solve specific cubic equations.

If taking cosine of one-third of the triple angle as a variable, given cosine of a triple angle, there are three solutions according to de Moivre’s formula. It is also true to find solutions for the cubic equations that are alike to the triple root identity.

Triple angle identity for sines function

Triple angle identity for sines function is used to represent sines of a triple angle in terms of sines of one-third of the triple angle. More specifically, sines of a triple angle equals to the difference between three multiples of sines of one-third of the triple angle and 4 multiples of sines of one-third of the triple angle to power three.

Triple angle identity for sines function is often used to simplify trigonometric expressions or solve specific cubic equations.

Given sines of a triple angle, there are three solutions for the cubic root of sines of one-third of the triple angle according to de Moivre’s formula. It is also true to find solutions for the cubic equations that are alike to the triple root identity.

Quotient Rule for Derivatives

The Quotient Rule says that the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator,

True Table for Logical Implication ⟹

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