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Maharashtra State BoardSSC (English Medium) 10th Standard

If cosec A – sin A = p and sec A – cos A = q, then prove that (p2q)23+(pq2)23 = 1 - Geometry Mathematics 2

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Question

If cosec A – sin A = p and sec A – cos A = q, then prove that `("p"^2"q")^(2/3) + ("pq"^2)^(2/3)` = 1

Sum
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Solution

cosec A – sin A = p   ......[Given]

∴ `1/"sin A" - sin "A"` = p

∴ `(1 - sin^2"A")/"sin A"` = p

∴ `(cos^2"A")/"sin A"` = p    ......`(i) [(because sin^2"A" + cos^2"A" = 1),(therefore 1 - sin^2"A" = cos^2"A")]`

sec A – cos A = q    ......[Given]

∴ `1/"cos A" - cos "A"` = q

∴ `(1 - cos^2"A")/"cos A"` = q

∴ `(sin^2"A")/"cos A"` = q   .....`[(because sin^2"A" + cos^2"A" = 1),(therefore 1 - cos^2"A" = sin^2"A")]`

L.H.S = `("p"^2"q")^(2/3) + ("pq"^2)^(2/3)`

= `[((cos^2"A")/(sin "A"))^2 ((sin^2"A")/(cos"A"))]^(2/3) + [((cos^2"A")/(sin "A"))((sin^2"A")/(cos"A"))^2]^(2/3)`  ......[From (i) and (ii)]

= `((cos^4"A")/(sin^2"A") xx (sin^2"A")/(cos"A"))^(2/3) + ((cos^2"A")/(sin"A") xx (sin^4"A")/(cos^2"A"))^(2/3)`

= `(cos^3"A")^(2/3) + (sin^3"A")^(2/3)`

= cos2A + sin2A

= 1

= R.H.S

∴ `("p"^2"q")^(2/3) + ("pq"^2)^(2/3)` = 1

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Chapter 6: Trigonometry - Q.5

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