Question

The value of $\frac{1}{2}\tan {585}^{\circ }+\sec (-{660}^{\circ })+\sin {405}^{\circ }\cot {510}^{\circ }-\text{cosec}(-{570}^{\circ })$ is

• A. $\frac{5}{2}-\sqrt{\frac{3}{2}}$
• B. $\frac{3}{2}-\sqrt{\frac{3}{2}}$
• C. $\frac{1}{2}+\sqrt{\frac{3}{2}}$
• D. $\frac{1}{2}-\sqrt{\frac{3}{2}}$

Answer 1

The correct option is D $\frac{1}{2}-\sqrt{\frac{3}{2}}$

$\frac{1}{2}\tan {585}^{\circ }+\sec (-{660}^{\circ })+\sin {405}^{\circ }\cot {510}^{\circ }-\text{cosec}(-{570}^{\circ })$

We know that,
$\tan {585}^{\circ }=-\tan (720-585{)}^{\circ }=-\tan {135}^{\circ }=1\sec (-{660}^{\circ })=\sec ({660}^{\circ })=\sec (720-660{)}^{\circ }=\sec {60}^{\circ }=2\sin {405}^{\circ }=\sin (360+45{)}^{\circ }=\sin {45}^{\circ }=\frac{1}{\sqrt{2}}\cot {510}^{\circ }=\cot (360+150{)}^{\circ }=\cot {150}^{\circ }=-\sqrt{3}\text{cosec}(-{570}^{\circ })=-\text{cosec}\left({570}^{\circ }\right)=\text{cosec}(720-570{)}^{\circ }=\text{cosec}({150}^{\circ })=2$

Now, putting the values in the expression, we get
$\frac{1}{2}\tan {585}^{\circ }+\sec (-{660}^{\circ })+\sin {405}^{\circ }\cot {510}^{\circ }-\text{cosec}(-{570}^{\circ })=\frac{1}{2}+2-\frac{1}{\sqrt{2}}\times \sqrt{3}-2=\frac{1}{2}+2-\sqrt{\frac{3}{2}}-2=\frac{1}{2}-\sqrt{\frac{3}{2}}$