Trigonometric Functions

Prove that:

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

Find the value of

(a) sin 18°

(b) cos 18°

(c) tan 18°

(d) sin 36°

(e) cos 36°

Find the values of the trigonometric functions

(a) cosec (−1410°)

(b) tan (19π / 3)

Prove that:

sin²(6x) − sin²(4x) = sin(2x) × sin(10x)

If α and β are the solutions of the equation a tan θ + b sec θ = c, then show that tan(α + β) = (2ac) / (a² − c²).

Solve:

sin2x - sin4x + sin6x = 0

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

Prove that:

(sin(x + y) / sin(x - y)) = (tan x + tan y) / (tan x - tan y)

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

Find the general solution:

sinx + sin3x+ sin5x = 0

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

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

Find the value of: tan(π/8)

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

Prove that: (sec8θ − 1) / (sec4θ − 1) = tan8θ / tan2θ

In triangle ABC, prove that:

tan((B − C)/2) = (b − c)/(b + c) × cot(A/2)

tan((C − A)/2) = (c − a)/(c + a) × cot(B/2)

tan((A − B)/2) = (a − b)/(a + b) × cot(C/2)

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

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

Show that:

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

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