Without using trigonometric tables, evaluate:

Without using trigonometric tables, prove that:

cos 81° - sin 9° = 0

Without using trigonometric tables, prove that:
tan 71° - cot 19° = 0

Without using trigonometric tables, prove that:
cosec 80° - sec 10° = 0

Without using trigonometric tables, prove that:
cosec² 72° - tan² 18° = 1

Without using trigonometric tables, prove that:
cos²75° + cos²15° =1

Without using trigonometric tables, prove that:
tan²66° - cot²24° = 0

Without using trigonometric tables, prove that:
sin²48^o + sin²42° = 1

Without using trigonometric tables, prove that:
cos²57° - sin²33° = 0

Without using trigonometric tables, prove that:
(sin 65° + cos 25°)(sin 65° - cos 25°) = 0

Without using trigonometric tables, prove that:

sin 53° cos 37° + cos 53° sin 37° = 1

cos 54° cos 36° - sin 54° sin 36° = 0

sec 70° sin 20° + cos 20° cosec 70° = 2

sin 35° sin 55° - cos 35° cos 55° = 0

(sin 72° + cos 18°)(sin 72° - cos 18°) = 0

tan 48° tan 23° tan 42° tan 67° = 1

Prove that:

Prove that:

sin θ cos (90° - θ) + sin (90° - θ ) cos θ = 1

cot θ tan (90° -θ) - sec (90° - θ)cosec θ +√3tan 12°tan 60°tan 78° = 2

Prove that:

tan 5° tan 25° tan 30° tan 65° tan 85° = 1/√3

cot 12° cot 38° cot 52° cot 60° cot 78° = 1/√3

cos 15° cos 35° cosec 55° cos 60° cosec 75° =1/2

cos 1° cos 2° cos 3° ... cos 180° = 0

Prove that:

sin (70°+θ) — cos (20° — θ) = 0

tan (55° — θ) — cot (35° + θ) = θ

cosec (67° + θ) — sec (23° — θ) = 0

cosec (65° + θ) — sec (25° —θ) — tan (55° — θ) + cot (35° + θ) =0

sin (50° +θ) — cos (40° — θ) + tan 1°tan 10°tan 80°tan 89° =1

Express each of the following in terms of T-ratios of angles lying between 0° and 45°:

sin 67° + cos 75°

Express each of the following in terms of T-ratios of angles lying between 0° and 45°:
cot 65° + tan 49°

Express each of the following in terms of T-ratios of angles lying between 0° and 45°:
sec 78° + cosec 56°

Express each of the following in terms of T-ratios of angles lying between 0° and 45°:
cosec 54° + sin 72°

If A, B and C are the angles of a ABC, prove that

If cos 2 θ = sin 4 θ, where 2θ and 4θ are acute angles, find the value of θ .

If sec 2A = cosec (A — 42°), where 2A is an acute angle, find the value of A.

If sin 3A = cos (A — 26°), where 3A is an acute angle, find the value of A.

If tan 2A = cot (A — 12°), where 2A is an acute angle, find the value of A.

If sec 4A = cosec (A — 15°), where 4A is an acute angle, find the value of A.

Prove that:

2/3 cosec² 58°-2/3 cot 58° tan 32° -5/3 tan 13° tan 37° tan 45° tan 53° tan 77° = —1.