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This essential trains for: SAT-II, AMC-10, AMC-12, AIME, Math Kangaroo 9-10.

The simplest trigonometric identity is the equivalent of the Pythagorean theorem. In a right angle triangle with hypotenuse of length c and legs of lengths a and b we have:

Divide this equation by c2:

and use the definitions of the trigonometric ratios to get:

where x is one of the acute angles of the triangle.

The sum and difference of arcs formulas can be derived geometrically. Draw right triangles ABC and ACD to share an edge like in the figure:

We want to compute:

Drop the perpendicular DP and then CN perpendicular onto DP.

The blue perpendicular will help us write the ratio for the sum of arcs:

Notice we can use the segment addition theorem:

Substitute:

Since:

the angles ∠CAB and ∠CDP have perpendicular sides and are, therefore, congruent.

As a result:

Rewrite the fraction:

Now work on the other fraction:

This can be written using trigonometric ratios:

Finally:

Now switch to the definitions of the trigonometric functions and use the parities of each function to derive:

Now substitute:

and use:

to get:

which, due to the parity of the cosine function, is the same as:

Using parity again:

Therefore, now we have all the sum and difference of arcs identities:

The double arc formulas can be derived by making x=y in the formulas for the sum of arcs:

The cosine of a double arc can be rewritten in three different ways by using the Pythagorean identity:

Using the definitions of the tangent and cotangent ratios:

Divide by cos2(x):

The half arc formulas can be derived by substituting:

in the double arc formulas for the cosine:

and, similarly: