Trigonometric Functions

Prove that:

tan(4x) = [4tan(x) × (1 − tan²(x))] / [1 − 6tan²(x) + tan⁴(x)]

Show that:

tan(3x) × tan(2x) × tan(x) = tan(3x) - tan(2x) - tan(x)

Find sin(x/2), cos(x/2) and tan(x/2), if tan(x) = −4/3, x in quadrant II

Prove: cos(π/5) × cos(2π/5) × cos(4π/5) × cos(8π/5) = −1/16

If: tanx = 3/4, π < x < 3π/2, find the value of: sin(x/2), cos(x/2), tan(x/2)

Prove: (cos 11° + sin 11°) / (cos 11° - sin 11°) = tan 56°

Prove that: tan 36° + tan 9° + tan 36° tan 9° = 1

Find sin(x/2), cos(x/2) and tan(x/2), if tan(x) = −4/3, x in quadrant II

Prove: (tan 5θ + tan 3θ) / (tan 5θ − tan 3θ) = 4 cos 2θ · cos 4θ

If tan α = m / (m + 1), tan β = 1 / (2m + 1), then find the value of α + β.

(a² + b²) / (a² + c²) = [(1 + cos(A − B)) × cos(C)] / [(1 + cos(A − C)) × cos(B)]

If: sinx = 3/5, cosy = −12/13, Where:x and y both lie in the second quadrant, Find the value of sin(x + y).

Prove that:

cot(x) × cot(2x) − cot(2x) × cot(3x) − cot(3x) × cot(x) = 1

asin(B − C) + bsin(C − A) + csin(A − B) = 0

Prove that:

cos(6x) = 32cos⁶(x) − 48cos⁴(x) + 18cos²(x) − 1

Prove that:

cos2x × cos(x/2) − cos3x × cos × (9x/2) = sin5x × sin × (5x/2)

Prove: sin 20° × sin 40° × sin 60° × sin 80° = 3/16

Find the value of: tan(π/8)

Solve: 2tan²x + sec²x = 2, for 0 ≤ x ≤ 2π

Show that: √(2 + √(2 + 2cos 4θ)) = 2cos θ