Prove that trignometry identity

Question

2 cos (45+x) cos (135-x) = -1+ sin 2x?

Answer 1

Recall the formula, cos A+cos B = 2 cos ((A+B)/2)os ((A-B)/2)

Here (A+B)/2=45+x A+B=90+2x ----- (1)

(A-B)/2=135-x A-B=270-2x ----- (2)

Solve equations (1) and (2)

Add equations (1) and (2)

2A=360 A=180

Substitute 180 for A in equation (1).

180+B=90+2x

B=-90+2x

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2 cos ((45+x)cos (135-x)=cos 180+cos(-90+2x)

                                    =-1+sin2x