Prove that sin 400 - cos 700 = √3 cos 800.

Solution:

Use sin A – sin B = 2 cos (A+B)/2 sin (A-B)/2

cos (90 – x) = sin x

So cos 70 = cos (90 – 20) = sin 20

sin 400 – cos 700 = sin 400 – sin 200

= 2 cos (40+20)/2 sin (40-20)/2

= 2 cos (60/2) sin (20/2)

= 2 cos 30 sin 100

= 2(√3/2) sin 100

= √3 cos 800 (since sin 10 = cos 80)

Hence proved.

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