Question
Download Solution PDFजर a cot θ + b cosec θ = p आणि b cot θ + a cosec θ = q तर p2 - q2 ______ च्या समान आहे.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFदिलेले 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
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