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now proving the last formula for a and B that is the fourth one no here we are having again cause a minus B now the formula is cos a cos B plus sine a sine B and for cos a plus B cos a cos B negative of sine a sine B now using these formulas again right hand side now using the formula cos a cos B sine a sine B in one bracket then cos a cos B opening the brackets changing the sine negative of Casa Cosby and positive of sine a sine B okay now these two are the same terms but a positive sign okay so we are left with twice of sign in/sign thing this is equal to left hand side left hand side is equal to right hand side hence twice of sine a sine B is equal to cos a minus B negative cos a plus B