Prove that: 1/[sin(x−a)cos(x−b)] = [cot(x−a) + tan(x−b)]/cos(a−b) Question Prove that: \[ \frac{1}{\sin(x-a)\cos(x-b)} = \frac{\cot(x-a)+\tan(x-b)} {\cos(a-b)} \] Proof R.H.S. \[ = \frac{\cot(x-a)+\tan(x-b)} {\cos(a-b)} \] \[ = \frac{ \frac{\cos(x-a)}{\sin(x-a)} + \frac{\sin(x-b)}{\cos(x-b)} } {\cos(a-b)} \] \[ = \frac{ \cos(x-a)\cos(x-b) + \sin(x-a)\sin(x-b) } {\cos(a-b)\sin(x-a)\cos(x-b)} \] Using \[ \cos C\cos D+\sin C\sin D = \cos(C-D) \] \[ = \frac{ \cos[(x-a)-(x-b)] […]

Prove that: 1/[sin(x−a)sin(x−b)] = [cot(x−a) − cot(x−b)]/sin(a−b) Question Prove that: \[ \frac{1}{\sin(x-a)\sin(x-b)} = \frac{\cot(x-a)-\cot(x-b)} {\sin(a-b)} \] Proof R.H.S. \[ = \frac{\cot(x-a)-\cot(x-b)} {\sin(a-b)} \] \[ = \frac{ \frac{\cos(x-a)}{\sin(x-a)} – \frac{\cos(x-b)}{\sin(x-b)} } {\sin(a-b)} \] \[ = \frac{ \cos(x-a)\sin(x-b) – \sin(x-a)\cos(x-b) } {\sin(a-b)\sin(x-a)\sin(x-b)} \] Using \[ \sin C\cos D-\cos C\sin D = \sin(C-D) \] \[ = \frac{ \sin[(x-b)-(x-a)]

If sinα + sinβ = a and cosα + cosβ = b, Show that sin(α+β) and cos(α+β) Question If \[ \sin\alpha+\sin\beta=a \] and \[ \cos\alpha+\cos\beta=b \] show that: \[ \text{(i)}\quad \sin(\alpha+\beta) = \frac{2ab}{a^2+b^2} \] \[ \text{(ii)}\quad \cos(\alpha+\beta) = \frac{b^2-a^2}{b^2+a^2} \] Proof Given, \[ \sin\alpha+\sin\beta=a \] \[ \cos\alpha+\cos\beta=b \] Squaring and adding, \[ (\sin\alpha+\sin\beta)^2 + (\cos\alpha+\cos\beta)^2

If 6cosx + 8sinx = 9, Find the Value of sin(α+β) Question If \( \alpha \) and \( \beta \) are two different values of \(x\) lying between \[ 0 \text{ and } 2\pi \] which satisfy: \[ 6\cos x+8\sin x=9 \] find the value of: \[ \sin(\alpha+\beta) \] Solution \[ 6\cos x+8\sin x=9 \]

If sin(α+β)=1 and sin(α−β)=1/2, Find tan(α+2β) and tan(2α+β) Question If \[ \sin(\alpha+\beta)=1 \] and \[ \sin(\alpha-\beta)=\frac12 \] where \[ 0\le \alpha,\beta \le \frac{\pi}{2} \] find: \[ \tan(\alpha+2\beta) \quad \text{and} \quad \tan(2\alpha+\beta) \] Solution Since \[ \sin(\alpha+\beta)=1 \] \[ \alpha+\beta=\frac{\pi}{2} \] Also, \[ \sin(\alpha-\beta)=\frac12 \] \[ \alpha-\beta=\frac{\pi}{6} \] Adding, \[ 2\alpha=\frac{\pi}{2}+\frac{\pi}{6} =\frac{2\pi}{3} \] \[ \alpha=\frac{\pi}{3} \]

If tan x + tan(x+π/3) + tan(x+2π/3) = 3, Prove that (3tanx − tan³x)/(1 − 3tan²x) = −1 Question If \[ \tan x+\tan\left(x+\frac{\pi}{3}\right)+\tan\left(x+\frac{2\pi}{3}\right)=3 \] prove that: \[ \frac{3\tan x-\tan^3x} {1-3\tan^2x} =-1 \] Proof Using \[ \tan3x = \frac{3\tan x-\tan^3x} {1-3\tan^2x} \] it is enough to prove that: \[ \tan3x=-1 \] Let \[ \tan x=t \]

If cos x = 8/17, Prove that cos(π/6+x) + cos(π/4−x) + cos(2π/3−x) = (23/17)[(√3−1)/2 + 1/√2] Question If \[ \cos x=\frac{8}{17} \] and \(x\) lies in the first quadrant, prove that: \[ \cos\left(\frac{\pi}{6}+x\right) + \cos\left(\frac{\pi}{4}-x\right) + \cos\left(\frac{2\pi}{3}-x\right) \] \[ = \frac{23}{17} \left( \frac{\sqrt3-1}{2} + \frac{1}{\sqrt2} \right) \] Proof Since \[ \cos x=\frac{8}{17} \] \[ \sin

If tan A + tan B = a and cot A + cot B = b, Prove that cot(A+B) = 1/a − 1/b Question If \[ \tan A+\tan B=a \] and \[ \cot A+\cot B=b \] prove that: \[ \cot(A+B)=\frac{1}{a}-\frac{1}{b} \] Proof Given, \[ \tan A+\tan B=a \] \[ \cot A+\cot B=b \] \[ \frac{1}{\tan

If cos A + sin B = m and sin A + cos B = n, Prove that 2sin(A+B) = m² + n² − 2 Question If \[ \cos A+\sin B=m \] and \[ \sin A+\cos B=n \] prove that: \[ 2\sin(A+B)=m^2+n^2-2 \] Proof Given, \[ \cos A+\sin B=m \] \[ \sin A+\cos B=n \]

If tan(A+B) = x and tan(A−B) = y, Find tan2A and tan2B Question If \[ \tan(A+B)=x \] and \[ \tan(A-B)=y \] find the values of: \[ \tan2A \quad \text{and} \quad \tan2B \] Solution Since, \[ 2A=(A+B)+(A-B) \] \[ \tan2A = \tan[(A+B)+(A-B)] \] \[ = \frac{\tan(A+B)+\tan(A-B)} {1-\tan(A+B)\tan(A-B)} \] \[ = \frac{x+y}{1-xy} \] Therefore, \[ \boxed{\tan2A=\frac{x+y}{1-xy}} \]