Question

Prove \[\dfrac{{1 + \cos A}}{{\sin A}} + \dfrac{{\sin A}}{{1 + \cos A}} = 2\cos ecA\]

Answer 1

Hint:
Here, we will solve the LHS of the given equation by taking the LCM of the denominators of the two given fractions. Then by using the suitable algebraic and trigonometric identities, and simplifying further we will prove the given equation.

Formula Used: We will use the following formulas:
1) \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
2) \[{\sin ^2}A + {\cos ^2}A = 1\]

Complete step by step solution:
We will first consider the left hand side of the equation.
LHS \[ = \dfrac{{1 + \cos A}}{{\sin A}} + \dfrac{{\sin A}}{{1 + \cos A}}\]
Now, taking LCM and solving further,
\[ \Rightarrow \] LHS \[ = \dfrac{{{{\left( {1 + \cos A} \right)}^2} + {{\sin }^2}A}}{{\sin A\left( {1 + \cos A} \right)}}\]
Using the identity \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\] in the numerator, we get
\[ \Rightarrow \] LHS \[ = \dfrac{{1 + 2\cos A + {{\cos }^2}A + {{\sin }^2}A}}{{\sin A\left( {1 + \cos A} \right)}}\]
Rewriting the above equation, we get
\[ \Rightarrow \] LHS \[ = \dfrac{{1 + 2\cos A + \left( {{{\cos }^2}A + {{\sin }^2}A} \right)}}{{\sin A\left( {1 + \cos A} \right)}}\]
We know that, \[{\sin ^2}A + {\cos ^2}A = 1\].
Hence, we get,
\[ \Rightarrow \] LHS \[ = \dfrac{{1 + 2\cos A + 1}}{{\sin A\left( {1 + \cos A} \right)}} = \dfrac{{2 + 2\cos A}}{{\sin A\left( {1 + \cos A} \right)}}\]
Taking 2 common from the numerator and then, cancelling out the same terms from the numerator as well as the denominator, we get,
\[ \Rightarrow \] LHS \[ = \dfrac{{2\left( {1 + \cos A} \right)}}{{\sin A\left( {1 + \cos A} \right)}} = \dfrac{2}{{\sin A}}\]
Using the trigonometric relations, we know that \[\cos ecA = \dfrac{1}{{\sin A}}\]
\[ \Rightarrow \] LHS \[ = \dfrac{2}{{\sin A}} = 2\cos ecA = \]RHS
Therefore, \[\dfrac{{1 + \cos A}}{{\sin A}} + \dfrac{{\sin A}}{{1 + \cos A}} = 2\cos ecA\].
Hence, proved

Additional information:
There are three primary functions of trigonometry; they are sine, tangent and cosine. Other three functions [cosecant, secant and cotangent] can be easily derived from the primary functions of trigonometry. Trigonometry is used in our real life for calculating height and distance also it is used in sectors like the department of marine, criminology etc.

Note:
Trigonometry is a branch of mathematics which helps us to study the relationship between the sides and the angles of a triangle. Trigonometric Ratios of a Particular angle are the ratios of the sides of a right angled triangle with respect to any of its acute angle. Trigonometric ratio and Pythagoras theorem are applicable only in the case of a Right Angle triangle.