Question

Prove sin x = 2 sinx/2 cosx/2.

Answer 1

Prove sin x = 2 sinx/2 cosx/2.

sinx can be written in the form of sin (x/2 + x/2)

= sin(x/2 + x/2)

= sinx/2 * cosx/2 + cosx/2* sinx/2

(by using sin(a+b)

= 2 sinx/2 * cosx/2.

Answer 2

sin x can be written as sin ( x/2 + x/2 ) .

We know that sin ( A + B ) = sin A cos B + cos A sin B .

⇒ sin x = sin x/2 × cos x/2 + cos x/2 × sin x/2

⇒ sin x = 2 sin x/2 cos x/2

Hence proved .

NOTE :

Important formulas :

sin ( A + B ) = sin A cos B + cos A sin B

sin ( A - B ) = sin A cos B - cos A sin B

cos ( A + B ) = cos A cos B - sin A sin B

cos ( A - B ) = cos A cos B + sin A sin B

From the above identities we can determine :

sin ( 2 x ) = sin x cos x + cos x sin x

⇒ sin 2 x = 2 sin x cos x

cos ( 2 x ) = cos x × cos x - sin x × sin x

⇒ cos ( 2 x ) = cos²x - sin²x

Another important thing :

In the first quadrant , all ratios are positive .

In the second quadrant , the ratio of sin is positive .

In the third quadrant , the ratio of tan is positive .

In the fourth quadrant the ratio of cos is positive .