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Prove that: sin (n + 1) x sin (n + 2)x +...

Prove that: `sin (n + 1) x sin (n + 2)x + cos (n + 1) x cos (n + 2) x = cos x`

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To prove the identity \( \sin(n+1)x \sin(n+2)x + \cos(n+1)x \cos(n+2)x = \cos x \), we will start from the left-hand side (LHS) and manipulate it to show that it equals the right-hand side (RHS). ### Step-by-Step Solution: **Step 1: Write down the left-hand side (LHS)** We start with the expression: \[ LHS = \sin(n+1)x \sin(n+2)x + \cos(n+1)x \cos(n+2)x ...
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Knowledge Check

  • The general value of x which satisfies the equation ( cos x + i sin x) ( cos 3 x + i sin 3 x) ( cos 5 x + i sin 5 x) . . . [ cos ( 2 n - 1) x + i sin ( 2 n - 1) x] = 1 is

    A
    `(r pi)/( n^(2))`
    B
    `(( r - 1) pi)/( n^(2))`
    C
    `((2 r + 1) pi)/( n^(3))`
    D
    ` (2 r pi)/( n^(2))`