Question

Prove that: \[\dfrac{{\cos \left( {90^\circ - A} \right) \cdot \sin \left( {90^\circ - A} \right)}}{{\tan \left( {90^\circ - A} \right)}} = {\sin ^2}A\]

Answer 1

Hint:
Here, we will first consider the left hand side of the equation and apply the formula of complementary angles. Then we will use the reciprocal trigonometric identity to simplify the equation further and prove that the left hand side of the given equation is equal to the right hand side.
Formula Used:
We will use the following formulas:
1) \[\cos \left( {90^\circ - A} \right) = \sin A\]
2) \[\sin \left( {90^\circ - A} \right) = \cos A\]
3) \[\tan \left( {90^\circ - A} \right) = \cot A\]
4) \[\cot A = \dfrac{{\cos A}}{{\sin A}}\]

Complete step by step solution:
In order to prove the given expression, we will solve the left hand side, i.e.
LHS \[ = \dfrac{{\cos \left( {90^\circ - A} \right) \cdot \sin \left( {90^\circ - A} \right)}}{{\tan \left( {90^\circ - A} \right)}}\]
Here, using the formulas, \[\cos \left( {90^\circ - A} \right) = \sin A\], \[\sin \left( {90^\circ - A} \right) = \cos A\] and \[\tan \left( {90^\circ - A} \right) = \cot A\], we get,
\[ \Rightarrow \] LHS \[ = \dfrac{{\sin A \cdot \cos A}}{{\cot A}}\]
Also, we know that, \[\cot A = \dfrac{{\cos A}}{{\sin A}}\], hence substituting this value in the denominator, we get,
\[ \Rightarrow \] LHS \[ = \dfrac{{\sin A \cdot \cos A}}{{\cos A}} \times \sin A\]
Cancelling out the same trigonometric functions from the numerator as well as the denominator, we get
\[ \Rightarrow \] LHS \[ = \sin A \times \sin A = {\sin ^2}A = \] RHS
Therefore,

\[\dfrac{{\cos \left( {90^\circ - A} \right) \cdot \sin \left( {90^\circ - A} \right)}}{{\tan \left( {90^\circ - A} \right)}} = {\sin ^2}A\]
Hence, proved

Additional Information:
Trigonometry is a branch of mathematics which helps us to study the relationship between the sides and the angles of a triangle. In practical life, trigonometry is used by cartographers to make maps. It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine and the cosine function.

Note:
We know that there are three basic ratios in trigonometric and they are sine, cosine and tangent ratio. While solving a trigonometric equation we need to keep in mind that every ratio needs to be converted in either sine or cosine ratio. This is because it makes the calculator less complicated. Every trigonometric function is periodic in nature, that is, the trigonometric function repeats itself after a certain interval.