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What is the value of the following?

\(\rm cosec\left(\dfrac{7\pi}{6}\right)sec\left(\dfrac{5\pi}{3}\right)\)


1. \(\dfrac{4}{3}\)
2. 4
3. -4
4. \(-\dfrac{4}{\sqrt3}\)

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Correct Answer - Option 3 : -4

Concept:

sec (360° -  θ) = sec θ;   

cosec (180° + θ) = - cosec θ

Calculation:

\(\rm cosec\left(\dfrac{7\pi}{6}\right)sec\left(\dfrac{5\pi}{3}\right)\)

= cosec (210) sec (300)

=  cosec(180 + 30) sec(360 - 60)

= - cosec 30 sec 60

= - 2 (2)

= - 4

 

sin (360° -  θ) = - sin θ            

cos (360° -  θ) = cos θ      

tan (360° -  θ= - tan θ      

cosec (360° -  θ) = - cosec θ        

sec (360° -  θ) = sec θ   

cot (360° -  θ= - cot θ

sin (180° + θ) = - sin θ

 

cos (180° + θ) = - cos θ

tan (180° + θ) = tan θ

cosec (180° + θ) = -cosec θ

sec (180° + θ) = - sec θ

cot (180° + θ) = cot θ

 

sin (90° - θ) = cos θ

cos (90° - θ) = sin θ

tan (90° - θ) = cot θ

cosec (90° - θ) = sec θ

sec (90° - θ) = cosec θ

cot (90° - θ) = tan θ

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