Hello, shadow85!
Prove: .
Consider: .
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The right side becomes: .
Soroban's method is completely fine. He focused on one specific part of the RHS and modified it so that the entire RHS is equal to the LHS. If you really on insist,
Whatever is in red is exactly what Soroban did. Does this conform to what you wanted?
Also, the reason why he split sin(3x) to sin(2x + x) is so that you can use the formula: . We (normally) aren't given the formula of sin(3x) in terms of sin x.