MQ8 --- Trigonometry

tanx
= (sina cosc - sinb sinc) / (cosa cosc - cosb sinc)
= (sina - sinb tanc) / (cosa - cosb tanc)
= (sina - tsinb ) / (cosa - tcosb) .............. (were t = tanc) tany
= (sina sinc - sinb cosc) / (cosa sinc - cosb cosc)
= (sina tanc - sinb) / (cosa tanc - cosb)
= (sinb - tsina) / (cosb - tcosa) .............. (were t = tanc)

∴ tan(x + y) = (tanx + tany) / (1 - tanx tany)
=
(sina - tsinb ) (cosb - tcosa) + (sinb - tsina) (cosa - tcosb)
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(cosa - tcosb) (cosb - tcosa) - (sina - tsinb) (sinb - tsina)
=
sina cosb - t(sina cosa + sinb cosb) + t²sinb cosa + sinb cosa - t(sina cosa + sinb cosb) + t²sina cosb
--------------------------------------------------------------------------------------------------------------------
cosa cosb - tcos²b - tcos²a + t²cosa cosb - sina sinb + tsin²b + tsin²a - t²sina sinb
=
(t² + 1)(sina cosb + sinb cosa) - 2t(sina cosa + sinb cosb)
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(t² + 1)(cosa cosb - sina sinb ) - t(cos²a - sin²a + cos²b - sin²b)
=
(t² + 1) sin(a + b) - t(sin2a + sin2b)
---------------------------------------------
(t² + 1) cos(a + b) - t(cos2a + cos2b)
=
(t² + 1) sin(a + b) - 2t sin(a + b) cos(a - b)
----------------------------------------------------
(t² + 1) cos(a + b) - 2t(cos(a + b) cos(a - b)
=
sin(a + b) [(t² + 1) - 2t cos(a - b)]
-----------------------------------------
cos(a + b) [(t² + 1) - 2t cos(a - b)]
=
tan (a + b)