If cos²(π/6 + x) − sin²(π/6 − x) = k cos 2x, Find k If cos²(π/6 + x) − sin²(π/6 − x) = k cos 2x, Find k Question: If \[ \cos^2\left(\frac{\pi}{6}+x\right) – \sin^2\left(\frac{\pi}{6}-x\right) = k\cos2x \] then \[ k= \] ……………………………………….. Solution Using the identities: \[ \cos^2\theta=\frac{1+\cos2\theta}{2} \] and \[ \sin^2\theta=\frac{1-\cos2\theta}{2} \] Therefore, \[ […]

If sin x cos y = 1/4 and 3 tan x = 4 tan y, Find sin(x − y) If sin x cos y = 1/4 and 3 tan x = 4 tan y, Find sin(x − y) Question: If \[ \sin x\cos y=\frac14 \] and \[ 3\tan x=4\tan y \] then \[ \sin(x-y) \]

Find the Value of cot(π/4 + x) cot(π/4 − x) Find the Value of cot(π/4 + x) cot(π/4 − x) Question: The value of \[ \cot\left(\frac{\pi}{4}+x\right) \cot\left(\frac{\pi}{4}-x\right) \] is …………………………………………….. Solution Using the identity: \[ \cot\theta=\frac{1}{\tan\theta} \] Therefore, \[ \cot\left(\frac{\pi}{4}+x\right) \cot\left(\frac{\pi}{4}-x\right) = \frac{1}{ \tan\left(\frac{\pi}{4}+x\right) \tan\left(\frac{\pi}{4}-x\right) } \] Now use: \[ \tan\left(\frac{\pi}{4}+x\right) = \frac{1+\tan x}{1-\tan x}

If x/cos θ = y/cos(θ − 2π/3) = z/cos(θ + 2π/3), Find x + y + z If x/cos θ = y/cos(θ − 2π/3) = z/cos(θ + 2π/3), Find x + y + z Question: If \[ \frac{x}{\cos\theta} = \frac{y}{\cos\left(\theta-\frac{2\pi}{3}\right)} = \frac{z}{\cos\left(\theta+\frac{2\pi}{3}\right)} \] then \[ x+y+z \] = …………………………………… Solution Let \[ \frac{x}{\cos\theta} = \frac{y}{\cos\left(\theta-\frac{2\pi}{3}\right)}

If cos(A − B) = 3/5 and tan A tan B = 2, Find sin A sin B If cos(A − B) = 3/5 and tan A tan B = 2, Find sin A sin B Question: If \[ \cos(A-B)=\frac{3}{5} \] and \[ \tan A\tan B=2 \] then \[ \sin A\sin B \] = …………………………………..

If A + B = π/4, Find (1 + tan A)(1 + tan B) If A + B = π/4, Find (1 + tan A)(1 + tan B) Question: If \[ A+B=\frac{\pi}{4} \] then \[ (1+\tan A)(1+\tan B) \] = …………………………………………….. Solution Using the tangent addition formula: \[ \tan(A+B) = \frac{\tan A+\tan B} {1-\tan A\tan

If A − B = π/4, Find (1 + tan A)(1 − tan B) If A − B = π/4, Find (1 + tan A)(1 − tan B) Question: If \[ A-B=\frac{\pi}{4} \] then \[ (1+\tan A)(1-\tan B) \] = …………………………………………….. Solution Using the tangent subtraction formula: \[ \tan(A-B) = \frac{\tan A-\tan B} {1+\tan A\tan

If sin θ + cos θ = 1, Find the Value of sin 2θ If sin θ + cos θ = 1, Find the Value of sin 2θ Question: If \[ \sin\theta+\cos\theta=1 \] then the value of \[ \sin2\theta \] is …………………………… Solution Given, \[ \sin\theta+\cos\theta=1 \] Squaring both sides, \[ (\sin\theta+\cos\theta)^2=1^2 \] \[ \sin^2\theta+\cos^2\theta+2\sin\theta\cos\theta=1

Find the Minimum Value of 4 cos x − 3 sin x + 7 Find the Minimum Value of 4 cos x − 3 sin x + 7 Question: The minimum value of \[ 4\cos x-3\sin x+7 \] is …………………………………………. Solution For an expression of the form \[ a\cos x+b\sin x \] the maximum value

Find the Maximum Value of 3 cos x + 4 sin x + 5 Find the Maximum Value of 3 cos x + 4 sin x + 5 Question: The maximum value of \[ 3\cos x+4\sin x+5 \] is ……………………………………….. Solution For an expression of the form \[ a\cos x+b\sin x \] the maximum value