Question

Brainly stars and moderators. A question for you all:.

sin x-sin 3x / cos x+cos 3x =tan 2x
Class 11
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Answer 1

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

$\longmapsto\tt{\dfrac{2sin\:\dfrac{x+3x}{2}\:cos\:\dfrac{x-3x}{2}}{2cos\:\dfrac{x+3x}{2}\:cos\:\dfrac{x-3y}{2}}}$

$\longmapsto\tt{\dfrac{2sin\:\dfrac{{\not{4x}}}{{\not{2}}}\:cos\:\dfrac{{\cancel{-2x}}}{{\not{2}}}}{2cos\:\dfrac{{\not{4x}}}{2}\:cos\:\dfrac{{\cancel{-2x}}}{{\not{2}}}}}$

$\longmapsto\tt{\dfrac{{\not{2}}\:sin\:2x\:{\cancel{cos-x}}}{{\not{2}}\:cos\:2x\:{\cancel{cos-x}}}}$

$\longmapsto\tt{\dfrac{sin\:2x}{cos\:2x}}$

$\longmapsto\tt\bf{tan\:2x}$

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

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Answer 2

Answer:

♧♧Perfect question:-

• Sin x +sin 3x/cosx+cos3x = tan 2x.

●Here given that,

• Sin x +Sin 3x / cos x +cos 3x

♧♧Here we should prove that,

• Sin x +sin3x / cos x +cos 3x =tan 2x.

♧♧Explanation :-

• Here we can use some of identities such as ,

1. Sinx +Sin y =2 sin x+y/2 cos x-y/2
2. cos x+ cosy = 2 cos x+y / 2 sin x-y/2

♧♧Now lets enter to your question,

2 sin x+3x /2 cos x -3x /2

=

2 cos x+3x /2 cos x- 3x /2

2 sin 4x/2 cos -2x/2

= 2 cos 4x/2 cos -2x/2

= 2 sin 2x cos -x

2 cos 2x cos -x

= sin 2x

cos 2x

= tan x .

Hence , here we should prove that,

sinx+sin 3x / cosx + cos 3x = tanx.

♧Therefore ,

LHS = RHS

♧Hence proved.

♧Hope it helps u mate .

♧Thank you.