Question

If $\cos 9\alpha = \sin \alpha$ and $9\alpha < 90^\circ$, then the value of $\tan 5\alpha$ is?
A) $\dfrac{1}{{\sqrt 3 }}$
B) $\sqrt 3$
C) $1$
D) $0$

Answer 1

We have given any equation involving sine and cosine functions. We can convert the $\sin \alpha$ by replacing it with $\cos (90^\circ - \alpha )$. Then we get an equation involving cosine only. Simplifying this we get the value of $\alpha$. Knowing the value of $\alpha$, we can find $5\alpha$ and thus $\tan 5\alpha$.

Formula used :
For any $\alpha$ we have, $\cos (90^\circ - \alpha ) = \sin \alpha$
$\tan 45^\circ = 1$

Complete step-by-step answer:
It is given that $\cos 9\alpha = \sin \alpha$.
We know that $\cos (90^\circ - \alpha ) = \sin \alpha$
This gives,
$\cos 9\alpha = \cos (90^\circ - \alpha )$
Comparing both sides we get,
$9\alpha = 90^\circ - \alpha$, since we have $9\alpha < 90^\circ$.
Adding $\alpha$ on both sides we get,
$9\alpha + \alpha = 90^\circ - \alpha + \alpha$
Simplifying we get,
$10\alpha = 90^\circ$
Dividing both sides by $10$ we get,
$\alpha = \dfrac{{90^\circ }}{{10}} = 9^\circ$
We need to find $\tan 5\alpha$.
We have, $\alpha = 9^\circ$
Multiplying both sides by $5$ we get,
$5\alpha = 5 \times 9^\circ = 45^\circ$
This gives,
$\tan 5\alpha = \tan 45^\circ$
We know that $\tan 45^\circ = 1$.
Substituting this, we have,
$\tan 5\alpha = 1$
$\therefore$ The answer is option C.

Note: We can do the problem in another way as well. Instead of converting the sine term to cosine term, we can convert cosine term to sine.
For that, we have $\sin (90^\circ - x) = \cos x$
Letting $x = 9\alpha$ we get,
$\sin (90^\circ - 9\alpha ) = \cos 9\alpha$
Given that $\cos 9\alpha = \sin \alpha$.
Substituting for $\cos 9\alpha$ using above result, we have
$\sin (90^\circ - 9\alpha ) = \sin \alpha$
So comparing both sides we get,
$90^\circ - 9\alpha = \alpha$
Adding $9\alpha$ on both sides we get,
$90^\circ - 9\alpha + 9\alpha = \alpha + 9\alpha$
Simplifying we get,
$90^\circ = 10\alpha$
Dividing both sides by $10$ we have,
$\alpha = 9^\circ$
Then we can proceed as above to get the answer.
Sine and cosine are periodic functions with period $360^\circ$. So, the same value can be taken for different angles. But here we have the condition $9\alpha < 90^\circ$. So we can restrict this to get a single value. Otherwise this is not possible.