Fundamental Trigonometry Identities

Useful suggestions for proving trigonometric identities:
1. Avoid aimless transformations. Any transformation that is made in one of the members should lead in some way to the form of the other.
2. Start with the more complicated member of the identity and transform it into the form of the simpler member.
3. Where possible, express different functions in terms of the same function.
4. It is often useful to express all functions in terms of sin and cos or in terms of tan or sec.
5. As a rule trigonometric functions of a double angle, a half angle, or the sums and differences of angles should be expressed in terms of functions of the single angle.
6. Simplify expressions by utilizing basic identies and combining like terms.
7. Simplify fractions. for example, transform complex fractions into simple fractions or divide the terms of a fraction by the common factors.

II. Solution of Trigonometric Equations

A. Useful suggestions for solving trigonometric equations

1. Simplify the equation by clearing fractions, removing parentheses, combining like terms, and removing radicals.
2. Express functions of a double angle, a half angle, or the sums and differences of angles in terms of functions of the single angle; then express the different functions of the single angle in terms of a single function of that angle.
3. Solve the resulting equation, whether it be linear or quadratic in nature, for all the values of the angle in the given domain.
4. Checks the results by substituting into the original equation.


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Sample Problems:
Verify:

sin x + cos x = cot x +1/ csc x...cot x = 1/tanx and csc x = 1/sin x
sin x + cos x = (1/tanx +1)/ (1/sin x)...multiply the left side of the equation by (sin x/1)
sin x + cos x = (sin x/tan x) + sin x...sin x/tan x =cos x so...
sin x + cos x = sin x + cos x


Verify:

sin x csc x=1...csc x= 1/sin x
sinx (1/sin x)= 1....multiply the two togather
sinx/sinx=1