Question

Given $\tan \theta =3\cot \theta$ then the value of the expression E = $\sin \theta +\cos \theta +\tan \theta +\sec \theta +cosec\theta +\cot \theta$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ where b. c and d are primes then the value of (a + b + c + d) equal

• A. 15
• B. 25
• C. 16
• D. 23

Answer 1

The correct option is A 15

$\tan \theta =3\cot \theta$

${\tan }^2\theta =3$

$\tan \theta =\sqrt{3}$

$\tan \theta =\tan 60$

$\theta =60$

$E=\sin \theta +\cos \theta +\tan \theta +\sec \theta +cosec\theta +\cot \theta$

$E=\sin 60+\cos 60+\tan 60+\sec 60+cosec60+\cot 60$

$E=\frac{\sqrt{3}}{2}+\frac{1}{2}+\sqrt{3}+2+\frac{2}{\sqrt{3}}+\frac{1}{\sqrt{3}}$

$E=\frac{3\sqrt{3}}{2}+\frac{5}{2}+\frac{3}{\sqrt{3}}$

$E=\frac{5\sqrt{3}}{2}+\frac{5}{2}$

$E=\frac{5+5\sqrt{3}}{2}$

Thus, $a=5,b=5,c=3,d=2$

$a+b+c+d=15$