Prove that sin α + sin β + sin γ − sin(α + β + γ) = 4 sin((α + β)/2) sin((β + γ)/2) sin((γ + α)/2) Prove that: \[ \sin\alpha+\sin\beta+\sin\gamma-\sin(\alpha+\beta+\gamma) \] \[ = 4\sin\frac{\alpha+\beta}{2} \sin\frac{\beta+\gamma}{2} \sin\frac{\gamma+\alpha}{2} \] Solution L.H.S. \[ = \sin\alpha+\sin\beta+\sin\gamma-\sin(\alpha+\beta+\gamma) \] Group first two terms and last two terms. Use identity: \[ […]

Prove that {sin(θ + Φ) – 2sinθ + sin(θ – Φ)}/{cos(θ + Φ) – 2cosθ + cos(θ – Φ)} = tan θ Prove that: \[ \frac{ \sin(\theta+\Phi)-2\sin\theta+\sin(\theta-\Phi) }{ \cos(\theta+\Phi)-2\cos\theta+\cos(\theta-\Phi) } = \tan\theta \] Solution L.H.S. \[ = \frac{ \sin(\theta+\Phi)-2\sin\theta+\sin(\theta-\Phi) }{ \cos(\theta+\Phi)-2\cos\theta+\cos(\theta-\Phi) } \] Use identities: \[ \sin C+\sin D = 2\sin\frac{C+D}{2}\cos\frac{C-D}{2} \] \[ \cos C+\cos

Prove that (sin A + 2 sin 3A + sin 5A)/(sin 3A + 2 sin 5A + sin 7A) = sin 3A/sin 5A Prove that: \[ \frac{ \sin A + 2\sin3A + \sin5A }{ \sin3A + 2\sin5A + \sin7A } = \frac{\sin3A}{\sin5A} \] Solution L.H.S. \[ = \frac{ \sin A + 2\sin3A + \sin5A }{

Prove that (sin A sin 2A + sin 3A sin 6A)/(sin A cos 2A + sin 3A cos 6A) = tan 5A Prove that: \[ \frac{ \sin A\sin2A+\sin3A\sin6A }{ \sin A\cos2A+\sin3A\cos6A } = \tan5A \] Solution L.H.S. \[ = \frac{ \sin A\sin2A+\sin3A\sin6A }{ \sin A\cos2A+\sin3A\cos6A } \] Use identity: \[ \sin C\sin D = \frac12[\cos(C-D)-\cos(C+D)]

Prove that (sin 3A cos 4A – sin A cos 2A)/(sin 4A sin A + cos 6A cos A) = tan 2A Prove that: \[ \frac{ \sin3A\cos4A-\sin A\cos2A }{ \sin4A\sin A+\cos6A\cos A } = \tan2A \] Solution L.H.S. \[ = \frac{ \sin3A\cos4A-\sin A\cos2A }{ \sin4A\sin A+\cos6A\cos A } \] Use identities: \[ \sin C\cos D

Prove that (sin 11A sin A + sin 7A sin 3A)/(cos 11A sin A + cos 7A sin 3A) = tan 8A Prove that: \[ \frac{ \sin11A\sin A+\sin7A\sin3A }{ \cos11A\sin A+\cos7A\sin3A } = \tan8A \] Solution L.H.S. \[ = \frac{ \sin11A\sin A+\sin7A\sin3A }{ \cos11A\sin A+\cos7A\sin3A } \] Use identities: \[ \sin C\sin D = \frac12[\cos(C-D)-\cos(C+D)]

Prove that (sin 5A cos 2A – sin 6A cos A)/(sin A sin 2A – cos 2A cos 3A) = tan A Prove that: \[ \frac{ \sin 5A \cos 2A – \sin 6A \cos A }{ \sin A \sin 2A – \cos 2A \cos 3A } = \tan A \] Solution L.H.S. \[ = \frac{

Prove that (sin 5A – sin 7A + sin 8A – sin 4A)/(cos 4A + cos 7A – cos 5A – cos 8A) = cot 6A Prove that: \[ \frac{ \sin 5A – \sin 7A + \sin 8A – \sin 4A }{ \cos 4A + \cos 7A – \cos 5A – \cos 8A } =

Prove that (sin 3A + sin 5A + sin 7A + sin 9A)/(cos 3A + cos 5A + cos 7A + cos 9A) = tan 6A Prove that: \[ \frac{\sin 3A + \sin 5A + \sin 7A + \sin 9A} {\cos 3A + \cos 5A + \cos 7A + \cos 9A} = \tan 6A \]

Prove that (cos 4A + cos 3A + cos 2A)/(sin 4A + sin 3A + sin 2A) = cot 3A Prove that: \[ \frac{\cos 4A + \cos 3A + \cos 2A} {\sin 4A + \sin 3A + \sin 2A} = \cot 3A \] Solution L.H.S. \[ = \frac{\cos 4A + \cos 3A + \cos 2A}