And sometimes that seems to be the only way to do the proof for some identities: work on the two sides until they meet in the middle, and then write something that looks like magic. I'm going to start with the lhs, work down to where the two sides meet, and then work up the ...
1. Simplify the equation by using trigonometric identities if necessary.2. Isolate the trigonometric function on one side of the equation.3. Use inverse trigonometric functions to find the angle measure.4. Check your solution by plugging it back into the original equation....
Plugging this definition of e raised to a complex power into the definitions of the hyperbolic trig functions in terms of e^x given above, one can easily obtain the identities sin(z) = -i sinh(iz) sinh(z) = i sin(-iz) = -i sin(iz) cos(z) = cosh(iz) cosh(z) = cos(-iz) ...