Prove that | 1 sin 3 x sin ^3 x 2 cos x sin 6 x sin ^3 2 x 4 cos ^2

Prove that | 1 sin 3 x sin ^3 x 2 cos x ...

Key Concepts

Determinants

A determinant is a scalar value that is computed from the elements of a square matrix and has many important properties, one of which is that if any set of rows or columns are linearly dependent, the determinant is zero. This property is widely used to prove identities or to check for linear dependence within the matrix structure.

Linear Dependence

Linear dependence of functions or vectors means that one of the rows or columns can be written as a linear combination of the others. When such a relationship exists in a matrix, it implies that the rows or columns do not add new information, leading to a determinant value of zero. Recognizing these dependencies is crucial for simplifying and proving equalities involving determinants.

Trigonometric Multiple Angle Formulas

Trigonometric multiple angle formulas, such as those for sin 3x or sin 6x, are identities that express functions of multiple angles in terms of basic sine or cosine functions. These identities, which often involve power expressions like sin^3 x, are instrumental in rewriting complex trigonometric expressions; such rewriting can reveal hidden linear relationships among the rows or columns in a trigonometric matrix.

Transcript

00:01 So in the given question we are told to prove that the determinant of the given matrix is equal to 0.

00:08 So this is the given determinant and we need to prove that the determinant of this matrix is equal to 0.

00:15 So what we are going to do first is we can take this matrix over here we have the determinant of the determinant of 1.

00:35 1, sine 3x, 2, cos x, 4, cos square x minus 1, sine 6x and sine 9x.

00:54 And in the last quarter, what we are going to do is we are going to use the formula that says sine 3x is equal to 3 sine x minus 4 sine cube x right and from this formula we can write sine cube x is equal to sine 3x sine 3x minus 3 sine x divided by minus 4 4 right we can write this or we can write it as 3 sine x minus sine 3x divided by so this is what we can write instead of sine cube x right using this formula we can write sine cube x is equal to 3 sine x minus sine 3x divided by 4 so let's write that over here instead of sine cube x so we will have sine 3 sine x minus sign 3x divided by 4.

02:23 In the case of sine cube 2x, we would have 3 times sine of 2x minus sign of 3 times 2x, which is 6x divided by 4.

02:39 And in the last position we would have 3 times sine 3x minus sine 9x divided by 4.

02:53 So this is what we would have.

02:56 Now let's put this in a column over here.

02:59 Let's highlight this formula that we just used and now in the next step what we are going to do is to take 1 by 4 outside.

03:11 Right so we can take 1 by 4 outside from this column over here so what we will have us 1 by 4 times 1 sine 3 x 3 sine x minus sine 3 x 2 cos x sine 6 x 3 of 2x minus sine 3 of 6x 4 coz square x minus 1 sine 9x and 3 sine 3x minus sine 3x minus sign of 9x so this is the step that we used and in this what we did is take 1 by 4 outside right and what is the next step so in the next step what we can do is we can add the columns 2 and 3 right so we are going to use the operation c3 is changed to c2 plus c3 so using this column operation what we would get is you would have 1 by 4 times and inside the determinant we would have 1 2 cos x 4 cos square 4 cause square x minus 1 and in the second column we have sine 3x 6x and sine 9 x and the last column we would have 3 sine 2x 3 sine 2x and 3 sine 3x right after the column operation this is the determinant that we have now what we do is we can further simplify this determinant by taking three outside as a common factor from this column so you would have 3 by 4 times 1 2 cos x 4 cos square x minus 1 sine 3x sine 6x, sine 9x...

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