cos 65°/sin 25° + cos 32°/sin 5

(i) cos 65°/sin 25° + cos 32°/sin 58° – sin 28° sec 62° + cosec² 30°

(ii) sec 29°/ cosec 61° + 2 cot 8° cot 17° cot 45° cot 73° cot 82° – 3 (sin² 38° + sin² 52°).

Answer :

(i) cos 65°/sin 25° + cos 32°/sin 58° – sin 28° sec 62° + cosec² 30°

= cos 65°/sin (90° – 65°) + cos 32°/sin (90°–32°) – sin 28° sec (90°–28°) + cosec² 30°

= cos 65°/cos 65° + cos 32°/cos 32° – sin 28° cosec 28° + cosec² 30°

cosec 30° = 2

= 1 + 1 – 1 + 4

= 5

(ii) sec 29°/cosec 61° + 2 cot8° cot17° cot45° cot73° cot82° – 3 (sin² 38° + sin² 52°)

= sec 29°/cosec (90° – 29°) + 2 cot8° cot17° cot45° cot (90°–17°) cot (90°–8°) – 3 [sin²38° + sin² (90°–38°)]

= sec 29°/sec 29° + (2 cot8° cot17° × 1 × tan17° tan8°) – 3 (sin² 38° + cos² 38°)

= 1 + (2 cot 8° tan 8° cot 17° tan 17° × 1) – (3 × 1)

cosec (90° – θ) = sec θ

⇒ cot (90° – θ) = tan θ

⇒ sin² θ + cos² θ = 1

= 1 + (2 × 1 × 1 × 1) – 3

= 1 + 2 – 3

= 0

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