Question

$\sin(n + 1) x \sin(n + 2) x + \cos(n + 1) x \cos(n + 2) x = \cos(x)$
prove it ❗☺​

Answer 1

hope this helps you☺️☺️☺️

Answer 2

To prove :

$\sf sin (n + 1)x . sin (n + 2)x + cos (n + 1)x . cos(n + 2 )x = cos (x)$

$\underline {\textbf{\large{Proof:}}}$

here, remember that,

$\sf cos ( A - B) = cosA.cosB + sinA .sinB$

LHS = $\sf sin (n + 1)x . sin (n + 2)x + cos (n + 1)x . cos(n + 2 )x$

= $\sf[ (cos (x ((n + 1) -(n +2)))]$

= $\sf[(cos(x (n + 1 - n - 2 )))]$

= $\sf[cos (x (-1))]$

= $\sf[cos (-x)]$

we know,

$\sf [ cos (-theta) = cos (theta)]$

therefore,

= $\sf cos (-x) = cos(x)$

= RHS

hence proved