Question

In any quadrilateral ABCD, prove that
(a) sin(A+B)+sin(C+D)=0
(b) cos(A+B)=cos(C+D)

Answer 1

Hint: Try to simplify the left-hand side of the equation given in the question by the application of the formula $\sin A+\sin B=2\sin \left( \dfrac{A+B}{2} \right)\cos \left( \dfrac{A-B}{2} \right)$ followed by the use of the fact that the sum of all the angles of a quadrilateral is $360{}^\circ$ .

Complete step-by-step answer:
Before starting with the solution, let us draw a diagram for better visualisation.

Now starting with the left-hand side of the first equation that is given in the question. According to the formula: $2\sin \left( \dfrac{X+Y}{2} \right)\cos \left( \dfrac{X-Y}{2} \right)=\sin \left( X \right)+sin\left( Y \right)$ , we get
$\sin \left( A+B \right)+\sin \left( C+D \right)$
$=2\sin \left( \dfrac{A+B+C+D}{2} \right)\cos \left( \dfrac{A+B-C-D}{2} \right)$
Now as ABCD is a quadrilateral, we can say:
$\angle A+\angle B+\angle C+\angle D=360{}^\circ$
So, substituting the value of A+B+C+D in our expression. On doing so, we get
$2\sin \left( \dfrac{360{}^\circ }{2} \right)\cos \left( \dfrac{A+B-C-D}{2} \right)$
$=2\sin 180{}^\circ \cos \left( \dfrac{A+B-C-D}{2} \right)$
We know $\sin 180{}^\circ =0$ . Using this in our expression, we get
$=2\times 0\times \cos \left( \dfrac{A+B-C-D}{2} \right)$
$=0$
Hence, we have proved the first equation.

Now starting with the left-hand side of the second equation that is given in the question. As ABCD is a quadrilateral, we can say:
$\angle A+\angle B+\angle C+\angle D=360{}^\circ$
$\angle A+\angle B=360{}^\circ -\left( \angle C+\angle D \right)$
So, substituting the value of A+B in our expression. On doing so, we get
cos(A+B)
$=\cos \left( 360{}^\circ -\left( C+D \right) \right)$
We know $\cos \left( 360{}^\circ -x \right)=\cos x$ . Using this in our expression, we get
$=\cos \left( 360{}^\circ -\left( C+D \right) \right)$
$=\cos \left( C+D \right)$
Hence, we have proved the second equation, as well.

Note: Be careful about the calculation and the signs while opening the brackets. Also, you need to learn the properties related to quadrilateral and other polygons, as they are used very often.