The Sum of Two Cosines or Two Sines

Suppose two right triangles, each with hypotenuse of length 1, are placed as in the following figure. The acute angle with measure A in the first triangle (yellow) is placed at the acute vertex opposite acute angle B in the second triangle (blue). Since the hypotenuse is of length 1 in each case we have a physical representations of sin A, sin B, cos A, and cos B as indicated in the figure.

By drawing segment XY we can complete a right triangle with legs of

sin A + sin B
cos A + cos B

Determine the length of XY in terms of the cosine and sine of some angle and then write trigonometric identities for sin A + sin B and for cos A + cos B.

Hint: Pay attention to the dotted line in the blue triangle that intersects the leg of the right triangle at an angle of measure A.

More Help

Reference: Kobayashi, Y. (1998) Trigonometric Identity: The Sume of Two Since or Two Cosines, The College Mathematics Journal, 29, 157.
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