First slide

Trigonometric Identities

Question

If $\sin x + cosec x + \tan y + \cot y = 4 where x and y \in [0, \frac{π}{2}],$ then tan $\frac{y}{2}$ is a root of the equation

Moderate

A

$α^{2} + 2 α + 1 = 0$
B

$α^{2} + 2 α - 1 = 0$
C

$2 α^{2} - 2 α - 1 = 0$
D

$α^{2} - α - 1 = 0$

Solution

It is true when $x = \frac{π}{2}, y = \frac{π}{4}$
$\begin{matrix} \tan y = 1 \Rightarrow \frac{2 \tan \frac{y}{2}}{1 - \tan^{2} \frac{y}{2}} = 1 \\ \Rightarrow \tan^{2} \frac{y}{2} + 2 \tan \frac{y}{2} = 1 \end{matrix}$

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