Question

10) Considerando x + y a expressão sen(x + y).sen(x - y) é equivalente a?
a) sen(x2 - y2)
b) sen(x2) + sen(72)
c) seny.senx + cosx.cosy
d) sen?x.sen?y
e) cos’x - cos x

Answer 1

(Ufsm 2002) Considerando x ≠ y, a expressão sen(x + y).sen(x - y) é equivalente a ?

$\begin{array}{lcl}\sin(x + y)\cdot\sin(x - y) &=& \sin(x + y)\cdot\sin(x - y) \\~\\&=& \left[\sin(x)\cos(y)+\sin(y)\cos(x)\right]\cdot\left[\sin(x)\cos(-y)+\sin(-y)\cos(x)\right]\\~\\&=&\left[\sin(x)\cos(y)+\sin(y)\cos(x)\right]\cdot\left[\sin(x)\cos(y)-\sin(y)\cos(x)\right]\\~\\&=&\sin^2(x)\cos^2(y)-\sin^2(y)\cos^2(x)\\~\\&=& \cos^2(y)\cdot[1-\cos^2(x)]-\sin^2(y)\cos^2(x)\\~\\&=& \cos^2(y)-\cos^2(y)\cos^2(x)-\sin^2(y)\cos^2(x) \\~\\&=& \cos^2(y)-\cos^2(x)\cdot[\cos^2(y)+\sin^2(y)]\end{array}$$\begin{array}{lcl}\sin(x + y)\cdot\sin(x - y) &=& \cos^2(y) -\cos^2(x)\end{array}$