Answer

Verified

382.5k+ views

**Hint:**The given question deals with basic simplification of trigonometric functions by using some of the simple trigonometric formulae such as $\sin \left( {{{90}^ \circ } - \theta } \right) = \cos \theta$. Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem and proving the result given to us.

**Complete step-by-step solution:**

In the given problem, we have to prove a trigonometric equality that can be further used in many questions and problems as a direct result and has wide ranging applications. For proving the desired result, we need to have a good grip over the basic trigonometric formulae and identities.

We are given that the angles A, B and C are the interior angles of a triangle.

Now, using the angle sum property of the triangle, we know that the sum of all the three interior angles of the triangle is ${180^ \circ }$. So, we get, $A + B + C = {180^ \circ }$.

So, we have, $A + B + C = {180^ \circ }$

Shifting the angle A from left side of the equation to right side of equation, we get,

$ \Rightarrow B + C = {180^ \circ } - A$

Now, dividing both sides of the equation by $2$, we get,

$ \Rightarrow \left( {\dfrac{{B + C}}{2}} \right) = \left( {\dfrac{{{{180}^ \circ } - A}}{2}} \right)$

Now, taking sine on both sides of the equation, we get,

$ \Rightarrow \sin \left( {\dfrac{{B + C}}{2}} \right) = \sin \left( {\dfrac{{{{180}^ \circ } - A}}{2}} \right)$

Simplifying the expression, we get,

$ \Rightarrow \sin \left( {\dfrac{{B + C}}{2}} \right) = \sin \left( {{{90}^ \circ } - \dfrac{A}{2}} \right)$

Now, we know that sine and cosine are the complementary trigonometric ratios. So, we have, $\sin \left( {{{90}^ \circ } - \theta } \right) = \cos \theta $.

$ \Rightarrow \sin \left( {\dfrac{{B + C}}{2}} \right) = \cos \left( {\dfrac{A}{2}} \right)$

So, we arrive at the equation that we needed to prove in the problem given to us.

Hence, Proved.

**Note:**Given problem deals with Trigonometric functions. For solving such problems, trigonometric formulae should be remembered by heart. Besides these simple trigonometric formulae, trigonometric identities are also of significant use in such types of questions where we have to simplify trigonometric expressions with help of basic knowledge of algebraic rules and operations.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which places in India experience sunrise first and class 9 social science CBSE

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE