Question

sin3A/sinA+cos3A/cosA=4cos2A
​

Answer 1

Step-by-step explanation:

Given :-

(Sin 3A / Sin A) + (Cos 3A / Cos A )

Required To Prove:-

Prove that (Sin 3A/Sin A)+(Cos 3A/Cos A )

= 4 Cos 2A

Solution:-

On taking LHS:

(Sin 3A / Sin A) + (Cos 3A / Cos A )

We know that

Sin 3A = 3 Sin A - 4 Sin³ A

Cos 3A = 4 Cos³ A - 3 Cos A

Now

(Sin 3A / Sin A)

=> (3 Sin A - 4 Sin³ A )/ Sin A

=> Sin A ( 3 - 4 Sin² A) / Sin A

=> 3 - 4 Sin² A

Sin 3 A / Sin A = 3 - 4 Sin² A --------------(1)

and

(Cos 3A / Cos A )

=> (4 Cos³ A - 3 Cos A) / Cos A

=> Cos A ( 4 Cos² A - 3) / Cos A

=> 4 Cos² A - 3

Cos 3 A / Cos A = 4 Cos² A - 3 ----------(2)

On adding (1)&(2)

=> (Sin 3A / Sin A) + (Cos 3A / Cos A )

=> ( 3 - 4 Sin² A ) + ( 4 Cos² A - 3 )

=> 3 - 4 Sin² A + 4 Cos² A - 3

=> -4 Sin² A + 4 Cos² A

=> 4 Cos² A - 4 Sin² A

=> 4 ( Cos² A - Sin² A )

We know that Cos² A - Sin² A = Cos 2A

=> 4 Cos 2A

=> RHS

=> LHS = RHS

Hence, Proved.

Answer:-

(Sin 3A / Sin A) + (Cos 3A / Cos A )

= 4 Cos 2A

Used formulae:-

→ Sin 3A = 3 Sin A - 4 Sin³ A

→ Cos 3A = 4 Cos³ A - 3 Cos A

→ Cos² A - Sin² A = Cos 2A