Question : $\triangle$XYZ is right angled at Y. If $\cos X=\frac{3}{5}$, then what is the value of $\operatorname{cosec}Z$?
Option 1: $\frac{3}{4}$
Option 2: $\frac{5}{3}$
Option 3: $\frac{4}{5}$
Option 4: $\frac{4}{3}$
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Correct Answer: $\frac{5}{3}$
Solution :
Given: $\triangle$XYZ is right-angled at Y and $\cos X=\frac{3}{5}$
Since $\triangle$XYZ is right-angled at Y,
So, X + Z = 90°
⇒ X = 90° – Z
Now, $\cos X=\frac{3}{5}$
⇒ $\cos(90°–Z)=\frac{3}{5}$
⇒ $\sin Z=\frac{3}{5}$
$\therefore\operatorname{cosec}Z=\frac{5}{3}$ [$\because$ $\operatorname{cosec}Z=\frac{1}{\sin Z}$]
Hence, the correct answer is $\frac{5}{3}$.
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