जर a cot θ + b cosec θ = p आणि b cot θ + a cosec θ = q तर p2 - q2 ______ च्या समान आहे.

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SSC CGL 2023 Tier-I Official Paper (Held On: 17 Jul 2023 Shift 3)
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  1. a2 + b2
  2. a2 - b2
  3. b2 - a2
  4. b - a

Answer (Detailed Solution Below)

Option 3 : b2 - a2
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Detailed Solution

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दिलेले आहे:

a cot θ + b cosec θ = p

b cot θ + a cosec θ = q

वापरलेले सूत्र:

Cosec2 θ - cot2 θ = 1

गणना:

a cot θ + b cosec θ = p

दोन्ही बाजूंचा वर्ग करूया

(a cot θ + b cosec θ)2 = (p)2

a2 cot2 θ  + b2 cosec2 θ  + 2 × ab cot θ × cosec θ = p2 ----- (1)

b cot θ + a cosec θ = q

दोन्ही बाजूंचा वर्ग करूया

(b cot θ + a cosec θ)2 = (q)2

b2 cot2 θ  + a2 cosec2 θ  + 2 × ab cot θ × cosec θ = q2 ----- (2)

समीकरण (1) आणि (2) वजा करणे

⇒ (p2 - q2) = a2 cot2 θ  + b2 cosec2 θ  + 2 × ab × cot θ × cosec θ - (b2 cot2 θ + a2 cosec2 θ  + 2 × ab × cot θ × cosec θ)

⇒ a2 cot2 θ - a2 cosec2 θ + b2 cosec2 θ - b2 cot2 θ 

⇒ a2 (cot2 θ - cosec2 θ) + b2 (cosec2 θ - cot2 θ) 

b2 - a2

∴ योग्य पर्याय 3 आहे.

 

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