Question : Using $\operatorname{cosec}(\alpha+\beta)=\frac{\sec \alpha imes \sec \beta imes \operatorname{cosec} \alpha imes \operatorname{cosec} \beta}{\sec \alpha imes \operatorname{cosec} \beta+\operatorname{cosec} \alpha imes \sec \beta}$, find the value of $\operatorname{cosec} 75°$.
Option 1: $\frac{\sqrt{6}+\sqrt{2}}{4}$
Option 2: $\frac{\sqrt{6}-\sqrt{2}}{4}$
Option 3: $\sqrt{6}-\sqrt{2}$
Option 4: $\sqrt{6}+\sqrt{2}$

Question

Question : Using $\operatorname{cosec}(\alpha+\beta)=\frac{\sec \alpha imes \sec \beta imes \operatorname{cosec} \alpha imes \operatorname{cosec} \beta}{\sec \alpha imes \operatorname{cosec} \beta+\operatorname{cosec} \alpha imes \sec \beta}$, find the value of $\operatorname{cosec} 75°$.

Option 1: $\frac{\sqrt{6}+\sqrt{2}}{4}$

Option 2: $\frac{\sqrt{6}-\sqrt{2}}{4}$

Option 3: $\sqrt{6}-\sqrt{2}$

Option 4: $\sqrt{6}+\sqrt{2}$

Team Careers360 · Accepted Answer

Correct Answer: $\sqrt{6}-\sqrt{2}$

Solution : Given: $\operatorname{cosec}(\alpha+\beta)=\frac{\sec \alpha imes \sec \beta imes \operatorname{cosec} \alpha imes \operatorname{cosec} \beta}{\sec \alpha imes \operatorname{cosec} \beta+\operatorname{cosec} \alpha imes \sec \beta}$
To find the value of $\operatorname{cosec} 75°$, take $\alpha=45°,\beta=30°$
Now putting their values in the equation we get,
$\operatorname{cosec}(45°+30°) = \frac{\sec 45° imes \sec 30° imes \operatorname{cosec} 45° imes \operatorname{cosec} 30°}{\sec 45° imes \operatorname{cosec} 30°+\operatorname{cosec} 45° imes \sec 30°}$
⇒ $\operatorname{cosec}(45°+30°) = \frac{\sqrt2 imes \frac{2}{\sqrt{3}} imes \sqrt2 imes 2 }{\sqrt 2×2+\sqrt2×\frac{2}{\sqrt{3}})}$
⇒ $\operatorname{cosec}(45°+30°) = \frac{\frac{2\sqrt2 imes 2\sqrt2}{\sqrt{3}}}{2×\sqrt{2}×(1+\frac{1}{\sqrt{3}})}$
⇒ $\operatorname{cosec}(45°+30°) = \frac{2×\sqrt{2}×(\sqrt{3}-1)}{(\sqrt{3}+1)×(\sqrt{3}-1)}$
⇒ $\operatorname{cosec}(45°+30°) = \frac{2×\sqrt{6}-2×\sqrt{2}}{2}$
⇒ $\operatorname{cosec}75°= \sqrt{6}-\sqrt{2}$
Hence, the correct answer is $(\sqrt{6}-\sqrt{2})$.

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Question : Using cosec(α+β)=sec⁡α×sec⁡β×cosecα×cosecβsec⁡α×cosecβ+cosecα×sec⁡β, find the value of cosec75°.Option 1: 6+24Option 2: 6−24Option 3: 6−2Option 4: 6+2

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