Question

number of solutions of cos^2(9x/2)+cos^2(7x/2)=1 in (0,π)​

Answer 1

Answer: 7

Step-by-step explanation:

cos^(9x/2) = 1 - cos^2(7x/2)

cos^2(9x/2) = sin^(7x/2)

cos(9x/2) = ± sin (7x/2)

cos(9x/2) = ± cos (π/2 - 7x/2)

now if cosA = ±cosB then, A = nπ±B

so, 9x/2 = nπ + 7x/2 or 9x/2 = nπ - 7x/2

    9x/2 -7x/2 = nπ or 9x/2 +7x/2 = nπ

         x = nπ or  8x = nπ ⇒ x =nπ/8

now since x ∈ (0,π). x can not take any value of nπ.

so x =nπ/8 and x ∈ (0,π)

means nπ/8 ∈ (0,π)

hence, n can be 1,2,3,4,5,6,7.

so number of solutions = 7

Answer 2

Given:

cos^2(9x/2) + cos^2(7x/2) = 1 in (0,π).

To Find:

The solution of the given question.

Solution:

cos^2(9x/2) + cos^2(7x/2) = 1.

It can be written as,

cos^(9x/2) = 1 - cos^2(7x/2)

We know that,

1 - cos^2 = sin^2.

So, it can be written as,

cos^2(9x/2) = sin^2(7x/2).

cos(9x/2) = ± sin (7x/2).

We know that,

sin x = cos (π/2 - x).

So,

cos(9x/2) = ± cos (π/2 - 7x/2).

now if cosA = ± cos B then, A = nπ ± B.

so, 9x/2 = nπ + 7x/2.

and 9x/2 = nπ - 7x/2.

9x/2 - 7x/2 = nπ.

and 9x/2 +7x/2 = nπ.

x = nπ.

and 8x = nπ.

x =nπ/8

Now since,

x ∈ (0,π). x can not take any value of nπ.

So,

x = nπ/8 and x ∈ (0,π).

= nπ/8 ∈ (0,π).

Hence, n can be 1,2,3,4,5,6,7.

So,

number of solutions = 7

Hence, the number of solutions is 7.