Question

value of (sin 2x +sin 5x - sin x)/(cos 2x + cos 5x + cos x)​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Answer:

The value of (sin 2x + sin 5x - sin x) ÷ (cos 2x + cos 5x + cos x)​ is tan2x.

Step-by-step calculation:

This sum involves the use of trigonometric formulas of multiple angles which are shown below as we calculate the answer:

$\frac{sin2x + sin5x- sinx}{cos2x + cos5x + cosx}$

$= \frac{sin2x + [2cos\frac{5x + x}{2} . sin\frac{5x -x}{2}] }{cos2x + [2cos\frac{5x + x}{2}. cos\frac{5x-x}{2}] }$
$\because sinC - sinD = 2cos(\frac{C+D}{2}).sin(\frac{C-D}{2})$, and,
  $cosC + cosD = 2cos(\frac{C+D}{2}).cos(\frac{C-D}{2})$

$= \frac{sin2x + 2cos3x.sin2x}{cos2x + 2cos3x.cos2x}$

$= \frac{sin2x.(1 + 2cos3x)}{cos2x.(1 + 2cos3x)}$

$= \frac{sin2x}{cos2x}$

$= tan2x$

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