Question

Prove that ( cosec A - sin A) ( sec A - cot A) sec^2A = tan A​

Answer 1

Step-by-step explanation:

L.H.S. = (cosec A – sin A)(sec A – cos A). sec2 A

= (1/sin A – sin A). (1/cos A – cos A). 1/cos2 A

= (1 – sin2 A/sin A) × (1 – cos2 A/cos A) × 1/cos2 a

= cos2 A/sin A × sin2 A/cos A × 1/cos2 A

[∵ (1 – sin2 A) = cos2 A]

[∵ 1 – cos2 A = sin2 A]

= sin A/cos A = tan A

= R.H.S.

Hence proved.

Answer 2

Correct Question :-

Prove that ( cosec A - sin A) ( sec A - cos A) sec^2A = tan A ?

Solution :-

→ ( cosec A - sin A) ( sec A - cos A) sec^2A = tan A

putting :-

• cosecA = (1/sinA)
• secA = (1/cosA)
• sec²A = (1/cos²A)

→ (1/sinA - sinA) * (1/cosA - cosA) * (1/cos²A) = tanA

→ {(1 - sin²A) /sinA} * {(1 - cos²A) /cosA} * (1/cos²A) = tanA

Putting :-

• (1 - sin²A) = cos²A
• (1 - cos²A) = sin²A

→ (cos²A/sinA) * (sin²A/cosA) * (1/cos²A) = tanA

→ (cos²A * sin²A) / (sinA * cos³A) = tanA

→ (sinA/cosA) = tanA

putting :-

• (sinA/cosA) = tanA

→ tanA = tanA (Proved).