prove this identity $\sin(x+y)\sin(x-y)=\sin^2 x - \sin^2 y$

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Solution 1

$\sin^2 x \cos^2 y-\cos^2 x \sin^2 y=\sin^2 x(1-\sin^2 y) -(1-\sin^2 x) \sin ^2 y$

$=\sin^2 x -\sin^2 x\sin^2y -\sin ^2 y + \sin^2x\sin^2y$

$=\sin^2 x - \sin ^2 y$

Solution 2

Use the identity $$\sin(x\pm y)=\sin x\cos y\pm \sin y\cos x$$ and we can get \begin{align*} LHS&=\sin(x+y)\sin(x-y)\\ &=(\sin x\cos y+\cos x\sin y)(\sin x\cos y-\cos x\sin y)\\ &=\sin^2x\cos^2y-\cos^2x\sin^2y \\ \end{align*}

See if you can take it from here using the identity $$\sin^2x+\cos^2x=1.$$

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Updated on July 21, 2022

Comments

  • dona12
    dona12 less than a minute

    prove this identity : $$\sin(x+y)\sin(x-y)=\sin^2 x - \sin^2 y$$ I tried solving it with additional formulas but I can't get the right answer. I get $$\sin^2 x \cos^2 y-\cos^2 x \sin^2 y$$

    • Ragnar
      Ragnar over 8 years
      Do you want a geometrical proof or just a proof using some common facts like $\sin^2+\cos^2=1$?
    • dona12
      dona12 over 8 years
      I think common facts
    • Martín-Blas Pérez Pinilla
      Martín-Blas Pérez Pinilla over 8 years
      Do you know the formulas for $\sin(x\pm y)$?