Show that: $\dfrac{\cos 30{}^\circ +\sin 60{}^\circ }{1+\sin 30{}^\circ +\cos 60{}^\circ }=\cos 30{}^\circ $ .

Hint: Try to simplify the left-hand side of the equation given in the question using the values of $\cos 30{}^\circ $ , $\sin 30{}^\circ $ and other required values. Just put the values and solve the expression to get the answer.

Complete step-by-step answer:


Before moving to the solution, let us discuss the periodicity of sine and cosine function, which we would be using in the solution. All the trigonometric ratios, including sine and cosine, are periodic functions. We can better understand this using the graph of sine and cosine.

First, let us start with the graph of sinx.



Next, let us see the graph of cosx.




Looking at both the graphs, we can say that the graphs are repeating after a fixed period i.e. $2{{\pi }^{c}}$ . So, we can say that the fundamental period of the cosine function and the sine function is $2{{\pi }^{c}}=360{}^\circ $

Now let us start with the left-hand side of the equation given in the question, we will try to simplify the expression given by putting the values $\cos 60{}^\circ =\dfrac{1}{2}$ . On doing so, we get

$\dfrac{\cos 30{}^\circ +\sin 60{}^\circ }{1+\sin 30{}^\circ +\dfrac{1}{2}}$

$=\dfrac{\cos 30{}^\circ +\sin 60{}^\circ }{\sin 30{}^\circ +\dfrac{3}{2}}$

Now we know that $\sin 30{}^\circ =\dfrac{1}{2}$ and $\cos 30{}^\circ =\dfrac{\sqrt{3}}{2}$ . So, if we put these values in our expression, we get

$=\dfrac{\dfrac{\sqrt{3}}{2}+\sin 60{}^\circ }{\dfrac{1}{2}+\dfrac{3}{2}}$

$=\dfrac{\dfrac{\sqrt{3}}{2}+\sin 60{}^\circ }{2}$

Finally, if we put the value $\sin 60{}^\circ =\dfrac{\sqrt{3}}{2}$ , we get

$=\dfrac{\dfrac{\sqrt{3}}{2}+\dfrac{\sqrt{3}}{2}}{2}$

$=\dfrac{\sqrt{3}}{2}$

And we know that $\dfrac{\sqrt{3}}{2}$ is the value of $\cos 30{}^\circ $ . So, we can say that the left-hand side of the equation given in the question is equal to the right- hand side of the same equation. Therefore, we have proved that $\dfrac{\cos 30{}^\circ +\sin 60{}^\circ }{1+\sin 30{}^\circ +\cos 60{}^\circ }=\cos 30{}^\circ $ .


Note: Be careful about the calculation and the signs of the formulas you use as the signs in the formulas are very confusing and are very important for solving the problems. Also, it would help if you remember the properties related to complementary angles and trigonometric ratios.