In trigonometry the reciprocals of $\sin$ and $\cos$ are called cosecant and secant. The reciprocal of tan is called cotangent. They are shortened as cosec, sec and cot. Hence, $\operatorname{cosec} x=\frac{1}{\sin x}, \sec x=\frac{1}{\cos x}, \cot x=\frac{1}{\tan x}$ Answer the following questions based on the above details. a $\sin x \times \operatorname{cosec} x=$ $\_\_\_\_$ b $\operatorname{cosec} 60^{\circ}=$ $\_\_\_\_$ c $\cot 45^{\circ}=$ $\_\_\_\_$ d What is $\sec 60^{\circ}-\operatorname{cosec} 30^{\circ}$ ? $$ \left(1,0, \frac{1}{2}, \frac{\sqrt{3}}{2}\right) $$

Explanation

To find the value of $\\sin x \\times \\operatorname{cosec} x$, we can use the definition of cosecant. $\\sin x \\times \\operatorname{cosec} x = \\sin x \\times (1 / \\sin x) = 1. To find the value of $\\operatorname{cosec} 60^{\\circ}$, we can use the definition of cosecant. $\\operatorname{cosec} 60^{\\circ} = 1 / \\sin 60^{\\circ} = 2. To find the value of $\\cot 45^{\\circ}$, we can use the definition of cotangent. $\\cot 45^{\\circ} = 1 / \\tan 45^{\\circ} = 1. To find the value of $\\sec 60^{\\circ} - \\operatorname{cosec} 30^{\\circ}$, we can use the definitions of secant and cosecant. $\\sec 60^{\\circ} = 1 / \\cos 60^{\\circ} = 2, \\operatorname{cosec} 30^{\\circ} = 1 / \\sin 30^{\\circ} = 2. Therefore, $\\sec 60^{\\circ} - \\operatorname{cosec} 30^{\\circ} = 2 - 2 = 0.

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