Question

Prove that 4 sin 23×sin 37×sin 83=cos 21

Answer 1

Sudut Relasi Sin (-a) = - sin a ; sin (180 - a) = sin a. Tan (-a) = - Tan a ; Tan (180 - a) = - Tan a. Sin (-150) = - sin 150 = - sin (180 - 30) = - sin 30 = - 1/2. Tan ...

Answer 2

Answer and Explanation:

To prove : $4\sin 23\times \sin 37\times \sin 83=\cos 21$

Proof :

Taking LHS,

$4\sin 23\times \sin 37\times \sin 83$

$=2\times 2\sin 23\times \sin 37\times \sin 83$

Applying formula,

$2\sin A\sin B=\cos(A-B)-\cos(A+B)$

$2\sin 23\times \sin 37=\cos(23-37)-\cos(23+37)$

$2\sin 23\times \sin 37=\cos(-14)-\cos(60)$

$2\sin 23\times \sin 37=\cos(14)-\cos(60)$

Substitute back,

$=2\times(\cos(14)-\cos(60))\times \sin 83$

$=2\cos(14)\sin 83-2\cos(60)\sin 83$

Again applying formula,

$2\cos A\sin B=\sin(A+B)-\sin(A-B)$ and $\cos 60=\frac{1}{2}$

Substitute above,

$=\sin (14+83)-\sin(14-83)-2\times \frac{1}{2}\times \sin 83$

$=\sin (97)-\sin(-69)- \sin 83$

$=\sin (90+7)+\sin(90-21)- \sin (90-7)$

$=\cos 7+\cos 21- \cos 7$

$=\cos 21$

$=RHS$

Hence proof , LHS=RHS.