A Prove: \ (\ SiN \ Alpha \ COS \ Beta = \ dfrac. 1} {2} {[\ SiN (\ Alpha + \ Beta) + \ SiN (\ alpha-\ Beta)] \)
Proof: Since \ [\ SiN (\ Alpha + \ Beta) = \ SiN \ Alpha \ COS \ Beta + \ COS \ Alpha \ SiN \ Beta \] \ [\ SiN (\ alpha-\ Beta) = \ SiN \ Alpha \ COS \ beta- \ cos \ alpha \ sin \ beta \] the left and right sides of the above two formulas were summed to give \ [\ sin (\ alpha + \ beta) + \ sin (\ alpha- \ beta) = 2 \ sin \ alpha \ cos \ beta \] i.e. \ [\ sin \ alpha \ cos \ beta = \ dfrac {1} {2} [\ sin (\ alpha + \ beta) + \ sin (\ alpha- \ beta)] \] with Li to give \ [\ COS \ Alpha \ SiN \ Beta = \ dfrac {. 1} {2} [\ SiN (\ Alpha + \ Beta) - \ SiN (\ alpha-\ Beta)] \] \ [\ COS \ Alpha \ COS \ Beta = \ dfrac {. 1} {2} [\ COS (\ Alpha + \ Beta) + \ COS (\ alpha-\ Beta)] \] \ [\ SiN \ Alpha \ SiN \ Beta = \ dfrac {. 1} {2} [\ cos (\ alpha + \ beta) - \ cos (\ alpha- \ beta)] \] Since the left side of the form as a product of the formula, and the right or in the form of a difference, it is referred to the above four equations plot and differential equations.
\(\quad\)
Second, verify: \ (\ SiN \ Theta + \ SiN \ varphi = 2 \ SiN \ dfrac {\ Theta + \ varphi} {2} \ COS \ dfrac {\ theta-\ varphi} {2} \)
Proof: There are a proof on a question \ [\ sin (\ alpha + \ beta) + \ sin (\ alpha- \ beta) = 2 \ sin \ alpha \ cos \ beta \] provided \ (\ alpha + \ beta = \ Theta, \ alpha-\ Beta = \ varphi \) . then \ [\ alpha = \ dfrac { \ theta + \ varphi} {2}, \ beta = \ dfrac {\ theta- \ varphi} {2} \] to \ (\ alpha, \ beta \) values into the above equation, i.e., to obtain \ [\ sin \ theta + \ sin \ varphi = 2 \ sin \ dfrac {\ theta + \ varphi} {2} \ cos \ dfrac {\ theta- \ varphi} {2} \] Similarly to obtain \ [\ sin \ theta- \ sin \ varphi = 2 \ cos \ dfrac {\ theta + \ varphi} {2} \ sin \ dfrac {\ theta- \ varphi} {2} \] \ [\ COS \ Theta + \ COS \ varphi = 2 \ COS \ dfrac {\ Theta + \ varphi} {2} \ COS \ dfrac {\ theta-\ varphi} {2} \] \ [\ COS \ theta- \ cos \ varphi = -2 \ sin \ dfrac {\ theta + \ varphi} {2} \ sin \ dfrac {\ theta- \ varphi} {2} \] we referred to the above four equations , and the difference of the plot formula.