Question

please give solution to above question of class 12 ch- derivatives​

Answer 1

Given :   y = Cos⁻¹{ ( 3 + 5 Cosx)/ (5 + 3 Cosx) }

To find : Prove that Cosx  = (4 - 5 y₁ )/3 y₁ where y₁  = dy/dx

Solution:

y = Cos⁻¹{ ( 3 + 5 Cosx)/ (5 + 3 Cosx) }

=> Cosy =  ( 3+5 Cosx)/ (5 +3 Cosx)

Differentiating wrtx  

-Siny . dy/dx  =    -5Sinx/ (5 + 3 Cosx)   + ( 3 + 5 Cosx) (-1/ (5 + 3 Cosx) ²)(-3Sinx)

=> -Siny . y₁  =  (-25Sinx  - 15SinxCosx +9Sinx +15SinxCosx)/(5 + 3 Cosx) ²

=> -Siny . y₁  =  (-16Sinx)/(5 + 3 Cosx) ²

=>  Siny . y₁  =  (16Sinx)/(5 + 3 Cosx) ²    . Eq1

Cosy =  ( 3+5 Cosx)/ (5 +3 Cosx)

=> Sin²y = 1  - (( 3+5 Cosx)/ (5 +3 Cosx) )²

=> Sin²y  = ( 1  + ( 3+5 Cosx)/ (5 +3 Cosx) ) (1  - ( 3+5 Cosx)/ (5 +3 Cosx) )

=>  Sin²y  = ( 8 + 8 Cosx )(2 - 2 Cosx) / (5 +3 Cosx) ²

=> Sin²y  =16 ( 1 +  Cosx )(1 -  Cosx) / (5 +3 Cosx) ²

=> Sin²y  =16 ( 1  -  Cos²x) / (5 +3 Cosx) ²

=> Sin²y  =16 ( Sin²x) / (5 +3 Cosx) ²

=> Siny = 4 Sinx/ (5 +3 Cosx)

Substituting in Eq 1

=> (4 Sinx/ (5 +3 Cosx)) . y₁   =  (16Sinx)/(5 + 3 Cosx) ²

=> (5 + 3 Cosx) . y₁  = 4

=> 5 y₁  + 3Cosxy₁  = 4

=> 3Cosxy₁  = 4  - 5 y₁

=> Cosx  = (4 - 5 y₁ )/3 y₁

QED

Hence proved

Learn More:

y = sin(tan-1 e^{-1}) dy/dx ज्ञात कीजिए

https://brainly.in/question/15286651

y = sin^{-1} 2x sqrt{1-x^{2}} dy/dx ज्ञात कीजिए

https://brainly.in/question/15286655