Let tan x = (2ab)/(a^{2} – b^{2}), where a > b > 0 and 0 < x < 90. Draw a right triangle with angle x and label two sides of the triangle to show tan x. Now find an expression for sin x.
sin x = (2ab)/c.
c^{2} | = (a^{2} – b^{2})^{2} + (2ab)^{2} |
= a^{4} – 2a^{2}b^{2} + b^{4} + 4a^{2}b^{2} | |
= a^{4} + 2a^{2}b^{2} + b^{4} | |
= (a^{2} + b^{2})^{2} |
So c = a^{2} + b^{2}, and sin x = (2ab)/(a^{2} + b^{2})