cos(a−b) cos(t+u) − cos(a+b) cos(t−u) = sin(u+a) sin(b−t) − sin(u−a) sin(b+t)

For the proof, just expand the following expression to show that it is zero:

cos(a−b) cos(t+u) − cos(a+b) cos(t−u) − sin(u+a) sin(b−t) + sin(u−a) sin(b+t)

=(cos a cos b + sin a sin b)(cos t cos u − sin t sin u)
−(cos a cos b − sin a sin b)(cos t cos u + sin t sin u)
−(sin u cos a + cos u sin a)(sin b cos t − cos b sin t)
+(sin u cos a − cos u sin a)(sin b cos t + cos b sin t)

= cos a cos b cos t cos u + sin a sin b cos t cos u − cos a cos b sin t sin u − sin a sin b sin t sin u
− cos a cos b cos t cos u + sin a sin b cos t cos u − cos a cos b sin t sin u + sin a sin b sin t sin u
− cos a sin b cos t sin u − sin a sin b cos t cos u + cos a cos b sin t sin u + sin a cos b sin t cos u
+ cos a sin b cos t sin u − sin a sin b cos t cos u + cos a cos b sin t sin u − sin a cos b sin t cos u

= 0

This identity is used in the proof of Urquhart's Theorem.

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