Prove that-cos x-cos y2-sin x-sin y2-4 sin 2x-y-2

We have L.H.S. = (cos x cos y)^2 + (sin x sin y)^2 = [ 2 sin (x+y/2) sin ( x y/2 ) ]^2 + [ 2 cos ( x+y/2) sin ( x y/2) ]^2 = 4 sin^2 ( x+y/2) sin^2 ( x ...

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Standard XII

Mathematics

Proof by mathematical induction

Prove that:co...

Question

Prove that:

$(cos x - cos y)^{2} + (sin x - sin y)^{2} = 4 {sin}^{2} \frac{x - y}{2}$

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Solution

We have L.H.S.

= $(cos x - cos y)^{2} + (sin x - sin y)^{2}$

= ${[- 2 sin (\frac{x + y}{2}) sin (\frac{x - y}{2})]}^{2}$ + ${[2 cos (\frac{x + y}{2}) sin (\frac{x - y}{2})]}^{2}$

= ${4 sin}^{2} (\frac{x + y}{2}) {sin}^{2} (\frac{x - y}{2}) + 4 {cos}^{2} (\frac{x + y}{2}) {sin}^{2} (\frac{x - y}{2})$

= ${4 sin}^{2} (\frac{x - y}{2})$ $[{sin}^{2} (\frac{x + y}{2}) + {cos}^{2} (\frac{x + y}{2})]$

= 4 ${sin}^{2} (\frac{x - y}{2})$ = R.H.S.

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$(cos x + cos y)^{2} + (sin x - sin y)^{2}$ = $4 {cos}^{2} (\frac{x + y}{2})$

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(cos x - cos y)^{2} + (sin x - sin y)^{2} = 4 {sin}^{2} \frac{x - y}{2}

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4 - 4 {sin}^{2} (\frac{x + y}{2}) = (sin x - sin y)^{2} + (cos x + cos y)^{2}

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(cos x + cos y)^{2} + (sin x - sin y)^{2} = 4 {cos}^{2} \frac{x + y}{2}

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Prove that: (cos x−cos y)2+(sin x−sin y)2=4sin2 x−y2

We have L.H.S. = (cos x−cos y)2+(sin x−sin y)2 = [−2sin(x+y2)sin(x−y2)]2 + [2cos(x+y2)sin(x−y2)]2 = 4 sin2(x+y2)sin2(x−y2)+4cos2(x+y2)sin2(x−y2) = 4 sin2(x−y2)[sin2(x+y2)+cos2(x+y2)] = 4 sin2(x−y2) = R.H.S.

Prove that:

$(cos x - cos y)^{2} + (sin x - sin y)^{2} = 4 {sin}^{2} \frac{x - y}{2}$