Question

Pls answer this Question with Explanations....​

Answer 1

Given:

Sin x/Sin y = 3

Cos x/Cos y = 1/2

Sin 2x/Sin 2y + Cos 2x/Cos 2y = a/b

To Find:

Value of b - a

Solution:

Sin x/Sin y = 3

Sin x = 3(Sin y) ...(1)

Squaring on both sides

Sin²x = 9Sin²y

And

Cos x/Cos y = 1/2

Cos x = 1/2(Cos y) ...(2)

Squaring on both sides

Cos²x = 1/4(Cos²y)

Adding equations (1) and (2)

Sin²x + Cos²x = 9Sin²y + 1/4(Cos²y)

Now as we know Sin²x + Cos²x = 1

1 = 9Sin²y + 1/4(Cos²y)

Multiplying 4 on both sides

4 = 36Sin²y + Cos²y

Now using identity Cos²y = 1 - Sin²y

4 = 36Sin²y + 1 - Sin²y

35Sin²y = 3

Sin²y = 3/35 ...(3)

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Sin 2x/Sin 2y + Cos 2x/Cos 2y = a/b

Now solving LHS of the above equation

Using identities

Sin 2x = 2SinxCosx

Cos 2x = 1 - 2 sin²x

2SinxCosx/ 2SinyCosy + 1 - 2Sin²x/ 1 - 2Sin²y

SinxCosx/ SinyCosy + 1 - 2Sin²x/ 1 - 2Sin²y

Now putting these values in above equation

Sin x = 3(Sin y) from equation (1)

Cos x = 1/2(Cos y) from equation (2)

Sin²y = 3/35 from equation (3)

(3×Siny×1/2Cosy)/SinyCosy + (1 - 2×9Sin²y)/(1-Sin²y)

Cancelling Siny and Cosy in numerator and denominator and putting value of Sin²y

3 × 1/2 + (1 - 2×9×3/35)/( 1 - 3/35)

3/2 + (1 - 54/35)/(1 - 3/35)

3/2 + (-19/35)/(29/35)

3/2 - 19/29

49/58 = LHS

Now comparing LHS and RHS we get

49/58 = a/b

a = 49

b = 58

Now evaluating value of b - a

b - a = 58 - 49 = 9

Hence, 9 is the required answer.