Question

Question for Brainly Teacher
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Answer 1

Step-by-step explanation:

Given:

$A=\left|\begin{array}{ccc}sinA&cosA&sinA+cosB\\sinB&cosA&sinB+cosB\\sinC&cosA&sinC+cosB\end{array}\right|$

To find:

Prove determinants is zero

Solution:

Expand the determinant from properties of determinants

$A=\left|\begin{array}{ccc}sinA&cosA&sinA\\sinB&cosA&sinB\\sinC&cosA&sinC\end{array}\right|$+$\left|\begin{array}{ccc}sinA&cosA&cosB\\sinB&cosA&cosB\\sinC&cosA&cosB\end{array}\right|$

Let $A_1=\left|\begin{array}{ccc}sinA&cosA&sinA\\sinB&cosA&sinB\\sinC&cosA&sinC\end{array}\right|$

$A_2=\left|\begin{array}{ccc}sinA&cosA&cosB\\sinB&cosA&cosB\\sinC&cosA&cosB\end{array}\right|$

A determinants is zero if two rows or two columns are identical.

in A1 determinant C1= C3

Thus,

A1=0

From A2

$A_2=\left|\begin{array}{ccc}sinA&cosA&cosB\\sinB&cosA&cosB\\sinC&cosA&cosB\end{array}\right|$

Take cos A common from C2 and cos B from C3

$A_2=cosA\:cosB\left|\begin{array}{ccc}sinA&1&1\\sinB&1&1\\sinC&1&1\end{array}\right|$

Again C2 and C3 are Identical ,thus

A2=0

So,

$A=A_1+A_2\\\\A=0+0\\\\A=0$

|A|=0

Hence proved.

Hope it helps you.