अपरः प्रत्युत्पन्नमतिः प्राह भविष्यदर्थे प्रमाणाभावात् कुत्र मया गन्तव्यम् । तदुत्पन्ने यथाकार्यं तदनुष्ठेयम्। तथा चोक्तम् - उत्पन्नामापदं यस्तु समाधत्ते स बुद्धिमान्।"
translate in English
Answers
Answer:
The roof and walls of the auditorium or cinema hall are generally covered with sound absorbent materials like draperies or compressed fibreboard to reduce reverberation. These materials reduce the formation of echoes by absorbing sound waves.
Answer:
Correct Question :
sin x + sin 3x / cos x + cos 3x = tan 2x
Given :
sin x + sin 3x / cos x + cos 3x
To Prove :
sin x-sin 3x / cos x+cos 3x =tan 2x
Solution :
Using Identities :
sin x + sin y = 2sin x+y/2 cos x-y/2
cos x + cos y = 2cos x+y/2 cos x-y/2
\longmapsto\tt{\dfrac{2sin\:\dfrac{x+3x}{2}\:cos\:\dfrac{x-3x}{2}}{2cos\:\dfrac{x+3x}{2}\:cos\:\dfrac{x-3y}{2}}}⟼
2cos
2
x+3x
cos
2
x−3y
2sin
2
x+3x
cos
2
x−3x
\longmapsto\tt{\dfrac{2sin\:\dfrac{{\not{4x}}}{{\not{2}}}\:cos\:\dfrac{{\cancel{-2x}}}{{\not{2}}}}{2cos\:\dfrac{{\not{4x}}}{2}\:cos\:\dfrac{{\cancel{-2x}}}{{\not{2}}}}}⟼
2cos
2
4x
cos
2
−2x
2sin
2
4x
cos
2
−2x
\longmapsto\tt{\dfrac{{\not{2}}\:sin\:2x\:{\cancel{cos-x}}}{{\not{2}}\:cos\:2x\:{\cancel{cos-x}}}}⟼
2cos2x
cos−x
2sin2x
cos−x
\longmapsto\tt{\dfrac{sin\:2x}{cos\:2x}}⟼
cos2x
sin2x
\longmapsto\tt\bf{tan\:2x}⟼tan2x
HENCE PROVED!
____________________
Some more Trigonometric Identities :
cos x + cos y = 2 cos x+y/2 cos x-y/2
sin x + sin y = 2 sin x+y/2 cos x-y/2
sin (-x) = -sin x
cos (-x) = cos x
cos (x-y) = cos x cos y + sin x sin y
sin (x-y) = sin x cos y - cos x sin y
____________________