Question

trignometry solve the following:-​

Answer 1

SOLUTION:

Trigonometric identities to remember:

• sin²θ + cos²θ = 1
• 1 - sin²θ = cos²θ
• 1 - cos²θ = sin²θ
• cosec²θ - cot²θ = 1
• cosec²θ - 1 = cot²θ
• 1 + cot²θ = cosec²θ
• sec²θ - tan²θ = 1
• sec²θ - 1 = tan²θ
• 1 + tan²θ = sec²θ
• cot²θ • tan²θ = 1
• sec²θ • cos²θ = 1
• sin²θ • cosec²θ = 1

Q: (cosec²θ - 1) tan²θ = 1

L.H.S = (cosec²θ - 1) tan²θ

= cot²θ • tan²θ

= 1 = R.H.S

Q: (1 + tan²θ) (1 - sin²θ) = 1

L.H.S = (1 + tan²θ) (1 - sin²θ)

= sec²θ • cos²θ

= 1 = R.H.S

Q: (1 - cos²θ) cosec²θ = 1

L.H.S = (1 - cos²θ) cosec²θ

= sin²θ • cosec²θ

= 1 = R.H.S

Q: (sec²θ - 1) cot²θ = 1

L.H.S = (sec²θ - 1) cot²θ

= tan²θ • cot²θ

= 1 = R.H.S

Answer 2

Explanation:

(Let theta = Ф)

(cosec²Ф-1)tan²Ф = 1

According to trigonometric identity:

cosec²Ф-1 = cot²Ф

Substitute:

→cot²Ф × tan²Ф

→ 1/tan²Ф × tan²Ф

→ 1

(1+tan²Ф)(1-sin²Ф) = 1

→sec²Ф × cos²Ф

→ 1/cos²Ф × cos²Ф

→ 1

(1 - cos²Ф) cosec²Ф = 1

→ sin²Ф × cosec²Ф

→ sin²Ф × 1/sin²Ф

→ 1

(sec²Ф - 1) cot²Ф = 1

→tan²Ф × cot²Ф

→ tan²Ф × 1/tan²Ф

→ 1

Hence proved for all the questions.