The value of cot 70^{circ} + 4 cos 70^{circ} | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $x = \cot 70^{\circ} + 4 \cos 70^{\circ}$.We can write this as $x = \frac{\cos 70^{\circ}}{\sin 70^{\circ}} + 4 \cos 70^{\circ} = \frac{\cos 7

Vedclass · Accepted Answer

$\sqrt{3}$

Vedclass · Answer

(B) Let $x = \cot 70^{\circ} + 4 \cos 70^{\circ}$.We can write this as $x = \frac{\cos 70^{\circ}}{\sin 70^{\circ}} + 4 \cos 70^{\circ} = \frac{\cos 70^{\circ} + 4 \sin 70^{\circ} \cos 70^{\circ}}{\sin 70^{\circ}}$.Using the identity $2 \sin 	heta \cos 	heta = \sin 2	heta$,we get $x = \frac{\cos 70^{\circ} + 2 \sin 140^{\circ}}{\sin 70^{\circ}}$.Since $\sin 140^{\circ} = \sin(180^{\circ} - 40^{\circ}) = \sin 40^{\circ}$,we have $x = \frac{\cos 70^{\circ} + 2 \sin 40^{\circ}}{\sin 70^{\circ}}$.Using $\cos 70^{\circ} = \sin 20^{\circ}$,$x = \frac{\sin 20^{\circ} + 2 \sin 40^{\circ}}{\sin 70^{\circ}} = \frac{\sin 20^{\circ} + \sin 40^{\circ} + \sin 40^{\circ}}{\sin 70^{\circ}}$.Using $\sin C + \sin D = 2 \sin \frac{C+D}{2} \cos \frac{C-D}{2}$,we get $\sin 20^{\circ} + \sin 40^{\circ} = 2 \sin 30^{\circ} \cos 10^{\circ} = 2 	imes \frac{1}{2} 	imes \cos 10^{\circ} = \cos 10^{\circ}$.So,$x = \frac{\cos 10^{\circ} + \sin 40^{\circ}}{\sin 70^{\circ}} = \frac{\sin 80^{\circ} + \sin 40^{\circ}}{\sin 70^{\circ}}$.Using $\sin C + \sin D = 2 \sin \frac{C+D}{2} \cos \frac{C-D}{2}$,we get $x = \frac{2 \sin 60^{\circ} \cos 20^{\circ}}{\sin 70^{\circ}} = \frac{2 	imes \frac{\sqrt{3}}{2} 	imes \cos 20^{\circ}}{\cos 20^{\circ}} = \sqrt{3}$.

The value of $\cot 70^{\circ} + 4 \cos 70^{\circ}$ is

Similar Questions

In a $\Delta ABC$,if the median,bisector,and altitude drawn from the vertex $A$ divide the angle at the vertex into four equal parts,then the angles of the $\Delta ABC$ are:

$\sinh^{-1}(2) + \cosh^{-1}(2) - \tanh^{-1}\left(\frac{2}{3}\right) + \coth^{-1}(-2) = $

If $\frac{\cos (\theta_1+\theta_2)}{\cos (\theta_1-\theta_2)}+\frac{\cos (\theta_3-\theta_4)}{\cos (\theta_3+\theta_4)}=0$,then $\cot \theta_1 \cdot \cot \theta_2 \cdot \cot \theta_3 \cdot \cot \theta_4=$

The value of the expression $\frac{2(\sin 1^{\circ} + \sin 2^{\circ} + \sin 3^{\circ} + \dots + \sin 89^{\circ})}{2(\cos 1^{\circ} + \cos 2^{\circ} + \dots + \cos 44^{\circ}) + 1}$ equals

If $0 < x < \pi$ and $\cos x + \sin x = \frac{1}{2}$,then $\tan x$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module