Question

If $\cos \left( {A + B} \right) = \alpha \cos A\cos B + \beta \sin A\sin B,$ then $\left( {\alpha ,\beta } \right) =$
A. $\left( { - 1, - 1} \right)$
B. $\left( { - 1,1} \right)$
C. $\left( {1, - 1} \right)$
D. $\left( {1,1} \right)$

Answer 1

Hint: In order to solve this type of question, first we will consider the given equation. Then, we will consider a trigonometric identity ($\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B$ ). Next, we will compare both the equations to find the value of $\alpha$ and $\beta$. Hence, we will get the required correct answer by finding the values of $\alpha$ and $\beta$.

Formula used: The trigonometric equation: $\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B$

Complete step-by-step solution:
We are given that,
$\cos \left( {A + B} \right) = \alpha \cos A\cos B + \beta \sin A\sin B$ …………………equation$\left( 1 \right)$
We know that the trigonometric equation for $\cos (A+B)$ is given as
$\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B$ …………………equation$\left( 2 \right)$
On comparing equations $\left( 1 \right)$ and $\left( 2 \right)$ we get,
$\alpha = 1,\;\beta = - 1$
Hence, $(\alpha ,\beta )=(1,-1)$
Hence, the correct option is C.

Note: The trigonometric equations should be remembered. Choose the suitable trigonometric identities and be very sure while simplifying them. A trigonometric equation is an equation involving one or more trigonometric ratios of unknown angles. A trigonometric equation can be solved in more than one way. This type of question requires the correct application of trigonometric rules to get the correct answer. Sometimes students get confused with the formula $\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B$ and $\cos \left( {A + B} \right) = \cos A\cos B + \sin A\sin B$.