Question

$\frac{sin5\theta +sin2\theta -sin\theta }{cos5\theta +2cos3\theta +2co{s}^2\theta +cos\theta }$ is equal to

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is B

We will combine two terms and apply the formula for sinA+sinB, or cosA+cosB... etc.. This is because we have 5$\theta$ and 3$\theta$ as the angles and there are no easy expansions of them.

Now we have to decide which all pairs we are going to combine. In the numerator, we will combine sin 5$\theta$ and sin$\theta$. This is because if we take 2$\theta$, after applying the transformation formula, we will have fractional angles (3.5$\theta$ and 1.5$\theta$)

In the denominator, we can either combine cos5$\theta$ and cos$\theta$ are cos5$\theta$ and cos3$\theta$. Both of them will simplify the expression. We will combine one cos3$\theta$ with cos5$\theta$ and the second one with cos$\theta$.

$\Rightarrow$ $\frac{sin5\theta -sin2\theta -sin\theta }{cos5\theta +2cos3\theta +2co{s}^2\theta +cos\theta }$

$\Rightarrow$ $\frac{(sin5\theta -sin\theta )+sin2\theta }{(cos5\theta +cos3\theta )+2co{s}^2\theta +(cos3\theta +cos\theta )}$

$\Rightarrow$ $\frac{2sin2\theta cos3\theta +sin2\theta }{2cos4\theta cos\theta +2co{s}^2\theta +2cos\theta cos2\theta }$

$\Rightarrow$ $\frac{sin2\theta (1+cos3\theta )}{2cos\theta (cos4\theta +cos\theta +cos2\theta )}$

(We don't have any more terms to combine in numerator. We can simplify it as 2$co{s}^2\frac{3\theta }{2}$.But it won't help us in further steps.)

= $\frac{sin2\theta }{2cos\theta }$ $\frac{(1+2cos3\theta )}{(2cos\theta cos3\theta +cos\theta )}$

= $\frac{sin2\theta }{2co{s}^2\theta }$ $\frac{(1+2cos3\theta )}{(1+2cos3\theta )}$

= $\frac{2sin\theta cos\theta }{2co{s}^2\theta }$

= tan$\theta$

key steps/concepts: (1) Combining 3terms (cosA$\mp$cosB,sinA$\mp$sinB )