If a cos 2x + b sin 2x = c Has Roots α and β, Prove tan α + tan β, tan α tan β and tan(α + β) If \[ a\cos2x+b\sin2x=c \] has roots \(\alpha\) and \(\beta\), prove that \[ (i)\quad \tan\alpha+\tan\beta=\frac{2b}{a+c} \] \[ (ii)\quad \tan\alpha\tan\beta=\frac{c-a}{c+a} \] \[ (iii)\quad \tan(\alpha+\beta)=\frac{b}{a} \] Question If \[ […]

If sin α = 4/5 and cos β = 5/13, Prove that cos((α − β)/2) = 8/√65 If \[ \sin\alpha=\frac{4}{5} \] and \[ \cos\beta=\frac{5}{13}, \] prove that \[ \cos\frac{\alpha-\beta}{2} = \frac{8}{\sqrt{65}} \] Question If \[ \sin\alpha=\frac{4}{5} \] and \[ \cos\beta=\frac{5}{13}, \] prove that \[ \cos\frac{\alpha-\beta}{2} = \frac{8}{\sqrt{65}}. \] Solution Given, \[ \sin\alpha=\frac{4}{5} \] Using \[

If cos α + cos β = 1/3 and sin α + sin β = 1/4, Prove that cos((α − β)/2) = ± 5/24 If \[ \cos\alpha+\cos\beta=\frac{1}{3} \] and \[ \sin\alpha+\sin\beta=\frac{1}{4}, \] prove that \[ \cos\frac{\alpha-\beta}{2} = \pm\frac{5}{24} \] Question If \[ \cos\alpha+\cos\beta=\frac{1}{3} \] and \[ \sin\alpha+\sin\beta=\frac{1}{4}, \] prove that \[ \cos\frac{\alpha-\beta}{2} = \pm\frac{5}{24}. \]

If sec(x + α) + sec(x − α) = 2 sec x, Prove that cos x = ± √2 cos(α/2) If \[ \sec(x+\alpha)+\sec(x-\alpha)=2\sec x, \] prove that \[ \cos x=\pm \sqrt{2}\cos\frac{\alpha}{2} \] Question If \[ \sec(x+\alpha)+\sec(x-\alpha)=2\sec x, \] prove that \[ \cos x=\pm \sqrt{2}\cos\frac{\alpha}{2}. \] Solution Given, \[ \sec(x+\alpha)+\sec(x-\alpha)=2\sec x \] Converting secant into cosine,

If cos x = (cos α + cos β)/(1 + cos α cos β), Prove that tan(x/2) = ± tan(α/2) tan(β/2) If \[ \cos x= \frac{\cos\alpha+\cos\beta} {1+\cos\alpha\cos\beta}, \] prove that \[ \tan\frac{x}{2} = \pm \tan\frac{\alpha}{2} \tan\frac{\beta}{2} \] Question If \[ \cos x= \frac{\cos\alpha+\cos\beta} {1+\cos\alpha\cos\beta}, \] prove that \[ \tan\frac{x}{2} = \pm \tan\frac{\alpha}{2} \tan\frac{\beta}{2}. \] Solution

If 2 tan(α/2) = tan(β/2), Prove that cos α = (3 + 5 cos β) / (5 + 3 cos β) If \[ 2\tan\frac{\alpha}{2}=\tan\frac{\beta}{2}, \] prove that \[ \cos\alpha= \frac{3+5\cos\beta}{5+3\cos\beta} \] Question If \[ 2\tan\frac{\alpha}{2} = \tan\frac{\beta}{2}, \] prove that \[ \cos\alpha = \frac{3+5\cos\beta}{5+3\cos\beta}. \] Solution Given, \[ 2\tan\frac{\alpha}{2} = \tan\frac{\beta}{2} \] Therefore, \[ \tan\frac{\alpha}{2}

If sin α + sin β = a and cos α + cos β = b, Prove that cos(α − β) = (a² + b² − 2)/2 If sin α + sin β = a and cos α + cos β = b, prove that \[ \cos(\alpha-\beta)=\frac{a^2+b^2-2}{2} \] Question If \[ \sin\alpha+\sin\beta=a \] and \[

If sin α + sin β = a and cos α + cos β = b, Prove that sin(α + β) = 2ab/(a² + b²) If sin α + sin β = a and cos α + cos β = b, prove that \[ \sin(\alpha+\beta)=\frac{2ab}{a^2+b^2} \] Question If \[ \sin\alpha+\sin\beta=a \] and \[ \cos\alpha+\cos\beta=b, \]

If 2 tan α = 3 tan β, Prove that tan(α − β) = sin 2β / (5 − cos 2β) If 2 tan α = 3 tan β, prove that \[ \tan(\alpha-\beta)=\frac{\sin2\beta}{5-\cos2\beta} \] Question If \[ 2\tan\alpha = 3\tan\beta, \] prove that \[ \tan(\alpha-\beta) = \frac{\sin2\beta}{5-\cos2\beta}. \] Solution Given, \[ 2\tan\alpha = 3\tan\beta \]

Prove that cos(π/65) cos(2π/65) cos(4π/65) cos(8π/65) cos(16π/65) cos(32π/65) = 1/64 Prove that: \[ \cos\frac{\pi}{65} \cos\frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65} = \frac{1}{64} \] Question Prove that \[ \cos\frac{\pi}{65} \cos\frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65} = \frac{1}{64} \] Solution Using the identity \[ 2\sin\theta\cos\theta=\sin2\theta \] Start with \[ \cos\frac{\pi}{65} \cos\frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65} \] Multiply and divide