If a cos 3 theta -3 a cos theta sin 2 theta - m and a sin 3 theta -3 a sin theta cos 2- n prove that m - n 2 - 3- m - n 2 - 3-2 a 2 - 3

m + n = acos3x + 3a cosx sin2x + asin3x + 3a cos2xsinx = a(sinx + cosx)3 (m + n)2/3 = a2/3 (sinx + cosx)2 m – n = acos3x + 3a cosx sin2x – asin3x – 3a cos2 ...

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Standard XII

Mathematics

Implicit Differentiation

If a cos 3 th...

Question

If a cos³ theta + 3a cos theta sin² theta = m and a sin ³theta + 3a sin theta cos ²= n prove that (m+n ) ^2/3+( m-n) ^2/3= 2a ^2/3

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Solution

m + n = acos³x + 3a cosx sin²x + asin³x + 3a cos²xsinx = a(sinx + cosx)³
(m + n)^2/3 = a^2/3 (sinx + cosx)²
m – n = acos³x + 3a cosx sin²x – asin³x – 3a cos²xsinx = a(cosx + sinx)³

(m – n)^2/3 = a^2/3 (cosx – sinx)²
(m + n)^2/3 + (m – n)^2/3 = a^2/3 (sinx + cosx)²+ a^2/3 (cosx – sinx)²=
a^2/3[(sinx + cosx)²+ (cosx – sinx)²] = 2a^2/3

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If a cos​​​​3 theta + 3a cos theta sin​​​​2 theta = m and a sin 3 theta + 3a sin theta cos 2 = n prove that (m+n ) 2/3 +( m-n) 2/3 = 2a 2/3

If a cos³ theta + 3a cos theta sin² theta = m and a sin ³theta + 3a sin theta cos ²= n prove that (m+n ) ^2/3+( m-n) ^2/3= 2a ^2/3