Question

If $\cot \left( {\alpha + \beta } \right) = 0$, then what is the value of $\sin \left( {\alpha + 2\beta } \right)$.
${\text{A}}{\text{. }}\sin \alpha \\\ {\text{B}}{\text{. }}\cos \alpha \\\ {\text{C}}{\text{. }}\sin \beta \\\ {\text{D}}{\text{. }}\cos \beta \\\$

Answer 1

Hint:Here, we will proceed by finding which angle will have its cotangent as zero using the general trigonometric table. Then, we will be equating the angles of the cotangents occurring on both sides of the equation.

Complete step-by-step answer:
Given, $\cot \left( {\alpha + \beta } \right) = 0{\text{ }} \to {\text{(1)}}$
According to the trigonometric table, $\cot \left( {\dfrac{\pi }{2}} \right) = 0{\text{ }} \to {\text{(2)}}$
By comparing equations (1) and (2), we get
$\cot \left( {\alpha + \beta } \right) = \cot \left( {\dfrac{\pi }{2}} \right){\text{ }} \to {\text{(3)}}$
As we know that if $\cot {\text{A}} = \cot {\text{B}}$, then A = B
Using the above concept in equation (3), we get
$\alpha + \beta = \dfrac{\pi }{2}{\text{ }} \to {\text{(4)}}$
Now, $\sin \left( {\alpha + 2\beta } \right) = \sin \left[ {\left( {\alpha + \beta } \right) + \beta } \right]$
By substituting equation (4) in the above equation, we get
$\Rightarrow \sin \left( {\alpha + 2\beta } \right) = \sin \left[ {\dfrac{\pi }{2} + \beta } \right]$
Using the formula $\sin \left( {\dfrac{\pi }{2} + \beta } \right) = \cos \beta$ in the above equation, we get
$\Rightarrow \sin \left( {\alpha + 2\beta } \right) = \cos \beta$
Therefore, the value of sine of angle $\left( {\alpha + 2\beta } \right)$ is equal to cosine of angle $\beta$.
Hence, option D is correct.

Note- In this particular problem, through the options we can easily see that the answer needs to be in terms of any one angle out of $\alpha$ and $\beta$. So, using the given equation we obtained a relation between the angles $\alpha$ and $\beta$. Then using that relation, the angle $\left( {\alpha + 2\beta } \right)$ is expressed only in terms of angle $\beta$.Students should remember trigonometric identities and trigonometric ratios for solving these types of problems.