Find the value of

To find: Value of $\operatorname{cosec}\left(-\frac{41 \pi}{4}\right)$

We have,

$\operatorname{cosec}\left(-\frac{41 \pi}{4}\right)=-\operatorname{cosec} \frac{41 \pi}{4}$

$[\because \operatorname{cosec}(-\theta)=-\operatorname{cosec} \theta]$

Putting π = 180°

$=-\operatorname{cosec} \frac{41 \times 180}{4}$

$=-\operatorname{cosec}\left[41 \times 45^{\circ}\right]$

$=-\operatorname{cosec}\left[1845^{\circ}\right]$

$=-\operatorname{cosec}\left[90^{\circ} \times 20+45^{\circ}\right]$

Clearly, $1845^{\circ}$ is in Ist Quadrant and the multiple of $90^{\circ}$ is even

$=-\operatorname{cosec} 45^{\circ}$

$=-\sqrt{2}\left[\because \operatorname{cosec} 45^{\circ}=\sqrt{2}\right]$