If `(sin^(4)theta)/(a)+(cos^(4)theta)/(b)=(1)/(a+b),` then which one of the following is incorrect?

Options

A. `(sin^(4)theta)/(a^(2))=(cos^(4)theta)/(b^(2))`
B. `(sin^(4)theta)/(b^(2))=(cos^(4)theta)/(a^(2))`
C. `(sin^(8)theta)/(a^(3))+(cos^(8)theta)/(b^(3))=(1)/((a+b)^(3))`
D. `sin^(4)theta=(a^(2))/((a+b)^(2))`

Answer

B

Explanation

We have,
`(sin^(4)theta)/(a)+(cos^(4)theta)/(b)=(1)/(a+b)`
`(a+b)((sin^(4)theta)/(a)+(cos^(4)theta)/(b))=(sin^(2)theta+cos^(2)theta)^(2)`
`impliessin^(4)theta+cos^(4)theta+b/asin^(4)theta+a/bcos^(4)theta`
`" "=(sin^(4)theta+cos^(4)theta+2sin^(2)thetacos^(2)theta)`
`(sqrt((b)/(a))sin^(2)theta-sqrt((a)/(b))cos^(2)theta)^(2)=0`
`impliessqrt((b)/(a))sin^(2)theta=sqrt((a)/(b))cos^(2)theta`
`impliesbsin^(2)theta=acos^(2)theta`
`implies(sin^(2)theta)/(a)=(cos^(2)theta)/(b)=(sin^(2)theta+cos^(2)theta)/(a+b)`
`impliessin^(2)theta=(a)/(a+b)and cos^(2)theta=(alpha)/(a+b)`
Clearly, these values satisfy options (a), (c ) and (d) only. Hence, option (b) is incorrect.