Question

Solve this trigonometry question

Prove it -


Cot 4x ( Sin 5x + Sin 3x ) = Cot x ( Sin 5x - Sin3x )​

Answer 1

$<b> <u> <i> Heya! </b> </u> </i>$

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$<b> <u> Trigonometry </b> </u>$

→ To prove :- Cot4x ( Sin 5x + Sin 3x ) = Cot x ( Sin 5x - Sin 3x)

★ Taking L.H.S.

=> Cot4x ( Sin 5x + Sin 3x)

$= > cot \: 4x \times 2 \: sin \: ( \frac{5x + 3x}{2} ) \: cos \: ( \frac{5x - 3x}{2} ) \\ \\ = > \frac{cos4x}{sin4x} \times 2 \: sin4x. \: cosx...............(eq.1)$

★Taking R.H.S.

$= > cot \: x \: (sin \: 5x - sin \:3 x) \\ \\ = > cot \: x \: \times 2sin( \frac{5x - 3x}{2} )cos \: (\frac{5x + 3x}{2} ) \\ \\ = > \frac{cos \: x}{sin \: x \: } \times 2sinx.cos4x................(eq.2)$

→ From Equations ( 1 ) and ( 2 )

$<b> L.H.S. = R.H.S. </b>$

$<u>Note of identities used :-</u>$

1.$<b> Cot x = Cos x / Sin x </b>$

2. $<b> Sin x + Sin y = 2 sin ( x + y /2 ) cos ( x - y/2)</b>$

3. $<b> Sin x - Sin y = 2 sin ( x - y /2 ) cos ( x +y/2)</b>$

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Answer 2

Cot4x ( Sin 5x + Sin 3x ) = Cot x ( Sin 5x - Sin 3x)

=Cot4x ( Sin 5x + Sin 3x)

= Cos 4x/Sin4x ( 2 Sin4xCosx)

= 2 × Cos 4x × Cos x

Other side of eq..

Cos x/Sinx ( 2 Cos4x Sinx)

= 2 × Cos 4x × Cos x

Thus both sides are equal so proofed