Educational

Prove that (sin x + sin 2x)/(1 + cos x + cos 2x) = tan x Prove that \[ \frac{\sin x+\sin 2x}{1+\cos x+\cos 2x}=\tan x \] Proof: \[ LHS=\frac{\sin x+\sin 2x}{1+\cos x+\cos 2x} \] Using the identity: \[ \sin 2x=2\sin x\cos x \] and \[ \cos 2x=2\cos^2x-1 \] Substituting these values: \[ LHS=\frac{\sin x+2\sin x\cos […]

Prove that (1 − cos 2x + sin 2x)/(1 + cos 2x + sin 2x) = tan x Prove that \[ \frac{1-\cos2x+\sin2x}{1+\cos2x+\sin2x}=\tan x \] Proof: \[ LHS=\frac{1-\cos2x+\sin2x}{1+\cos2x+\sin2x} \] Using the identities: \[ 1-\cos2x=2\sin^2x \] \[ 1+\cos2x=2\cos^2x \] \[ \sin2x=2\sin x\cos x \] Substituting these values: \[ LHS=\frac{2\sin^2x+2\sin x\cos x}{2\cos^2x+2\sin x\cos x} \] Taking common factors:

Prove that √(2 + √(2 + 2cos 4x)) = 2cos x Prove that \[ \sqrt{2+\sqrt{2+2\cos4x}}=2\cos x \] for \[ 0

Prove that sin 2x /(1 + cos 2x) = tan x Prove that \[ \frac{\sin 2x}{1+\cos 2x}=\tan x \] Proof: \[ LHS=\frac{\sin 2x}{1+\cos 2x} \] Using the identities: \[ \sin 2x=2\sin x\cos x \] \[ 1+\cos 2x=2\cos^2 x \] Substituting these values: \[ LHS=\frac{2\sin x\cos x}{2\cos^2 x} \] \[ =\frac{\sin x\cos x}{\cos^2 x} \] \[

Prove that sin 2x /(1 – cos 2x) = cot x Prove that \[ \frac{\sin 2x}{1-\cos 2x}=\cot x \] Proof: \[ LHS=\frac{\sin 2x}{1-\cos 2x} \] Using the identities: \[ \sin 2x=2\sin x\cos x \] \[ 1-\cos 2x=2\sin^2 x \] Substituting these values: \[ LHS=\frac{2\sin x\cos x}{2\sin^2 x} \] \[ =\frac{\sin x\cos x}{\sin^2 x} \] \[

Prove that √((1 − cos 2x)/(1 + cos 2x)) = tan x Prove that \[ \sqrt{\frac{1-\cos 2x}{1+\cos 2x}}=\tan x \] Proof: \[ LHS=\sqrt{\frac{1-\cos 2x}{1+\cos 2x}} \] Using the identities: \[ 1-\cos 2x=2\sin^2 x \] \[ 1+\cos 2x=2\cos^2 x \] Substituting these values: \[ LHS=\sqrt{\frac{2\sin^2 x}{2\cos^2 x}} \] \[ =\sqrt{\frac{\sin^2 x}{\cos^2 x}} \] \[ =\sqrt{\tan^2 x}

If m sin θ = n sin(θ + 2α), prove that tan(θ + α) cot α = (m + n)/(m − n) If \[ m\sin\theta=n\sin(\theta+2\alpha) \] prove that \[ \tan(\theta+\alpha)\cot\alpha = \frac{m+n}{m-n} \] Solution Given: \[ m\sin\theta=n\sin(\theta+2\alpha) \] Use identity: \[ \sin(A+B)=\sin A\cos B+\cos A\sin B \] \[ m\sin\theta = n[\sin\theta\cos2\alpha+\cos\theta\sin2\alpha] \] Bring terms containing

If x cos θ = y cos(θ + 2π/3) = z cos(θ + 4π/3), prove that xy + yz + zx = 0 If \[ x\cos\theta = y\cos\left(\theta+\frac{2\pi}{3}\right) = z\cos\left(\theta+\frac{4\pi}{3}\right) \] prove that \[ xy+yz+zx=0 \] Solution Let \[ x\cos\theta = y\cos\left(\theta+\frac{2\pi}{3}\right) = z\cos\left(\theta+\frac{4\pi}{3}\right) =k \] Then \[ x=\frac{k}{\cos\theta} \] \[ y=\frac{k}{\cos\left(\theta+\frac{2\pi}{3}\right)} \] \[ z=\frac{k}{\cos\left(\theta+\frac{4\pi}{3}\right)}

If cos(A + B) sin(C − D) = cos(A − B) sin(C + D), prove that tan A tan B tan C + tan D = 0 If \[ \cos(A+B)\sin(C-D) = \cos(A-B)\sin(C+D) \] prove that \[ \tan A\tan B\tan C+\tan D=0 \] Solution Given: \[ \cos(A+B)\sin(C-D) = \cos(A-B)\sin(C+D) \] Use identities: \[ \cos(X+Y)=\cos X\cos Y-\sin

If y sin Φ = x sin(2θ + Φ), prove that (x + y) cot(θ + Φ) = (y − x) cot θ If \[ y\sin\Phi=x\sin(2\theta+\Phi) \] prove that \[ (x+y)\cot(\theta+\Phi)=(y-x)\cot\theta \] Solution Given: \[ y\sin\Phi=x\sin(2\theta+\Phi) \] Use identity: \[ \sin(A+B)=\sin A\cos B+\cos A\sin B \] \[ y\sin\Phi = x[\sin2\theta\cos\Phi+\cos2\theta\sin\Phi] \] Use identities: \[ \sin2\theta=2\sin\theta\cos\theta