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If $a \sin^{3} x + b \cos^{3} x = sinxcosx$ and $asinx = bcosx$ . If $a, b, sinx$ and $cosx$ all are non zero real numbers, then $a^{2} + b^{2} =$ ?

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0

1

2

3

answer is B.

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Detailed Solution

Given the equations,

a \sin^{3} x + b \cos^{3} x = sinx cosx . . . . . . . . . . . (1)

asinx = bcosx . . . . . . . . . (2)

Consider equation

(1)

a \sin^{3} x + b \cos^{3} x = sinx cosx

a sinx . \sin^{2} x + b \cos^{3} x = sinx cosx

But,

asinx = bcosx .

So,

bcosx \sin^{2} x + b \cos^{3} x = sinx cosx

Taking

bcosx

in LHS,

bcosx (\sin^{2} x + \cos^{2} x) = sinx cosx

Solving this, we get

b = sinx

.
Then,

a = \frac{bcosx}{sinx} = cosx

.
Therefore,

a^{2} + b^{2} = \sin^{2} x + \cos^{2} x = 1

.
Hence, the correct option is (2).

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