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  • In ABC let L, M, N be the feet of the altitudes. Then  <img alt="\sin \angle MLN + \sin \angle LMN + \sin

In \DeltaABC let L, M, N be the feet of the altitudes. Then 

\sin \angle MLN + \sin \angle LMN + \sin \angle MNL equals to

Option: 1

4 \sin A \sin B \sin C


Option: 2

4 \cos A \cos B \cos C


Option: 3

\tan A + \tan B + \tan C


Option: 4

none 


Answers (1)

best_answer

As we have learnt in

 

Trigonometric Ratios of Functions -

\sin \Theta = \frac{Opp}{Hyp}

\cos \Theta = \frac{Base}{Hyp}

\tan \Theta = \frac{Opp}{Base}

- wherein

Trigonometric Ratios of Functions

 

 

Using properties of pedal triangle,

                we have         \sin \angle MLN + \sin \angle LMN + \sin \angle MNL

 MLN = 180^{o} - 2A

LMN = 180 \degree - 2B \\\\ \angle MNL = 180 \degree - 2 C

                Hence the required sum    

                                = \sin2A + \sin2B + \sin2C = 4 \sin A \sin B \sin C

 

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mansi

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