Question

How do you find the value of $\cos 70\sin 48-\cos 48\sin 70$ using the sum and difference formulas?

Answer 1

Hint: We start solving the problem by equating the given term to a variable. We then recall the sum and difference formula of the sine function as $\sin \left( A-B \right)=\sin A\cos B-\cos A\sin B$. We then compare the given term with formula to find the values of A and B to proceed through the problem. We then make the necessary calculations and then make use of the fact that $\sin \left( -x \right)=-\sin x$ which gives the required answer for the given problem.

Complete step by step answer:
According to the problem, we are asked to find the value of $\cos 70\sin 48-\cos 48\sin 70$ using the sum and difference formulas.
Let us assume $d=\cos 70\sin 48-\cos 48\sin 70$ ---(1)
We know that the sum and difference formula of sine function is defined as $\sin \left( A-B \right)=\sin A\cos B-\cos A\sin B$. Let us use this result in equation (1). On comparing $\cos 70\sin 48-\cos 48\sin 70$ with $\sin A\cos B-\cos A\sin B$, we get $A=48$ and $B=70$.
$\Rightarrow d=\sin \left( 48-70 \right)$.
$\Rightarrow d=\sin \left( -22 \right)$ ---(2).
We know that $\sin \left( -x \right)=-\sin x$. Let us use this result in equation (2).
$\Rightarrow d=-\sin 22$.
So, we have found the value of the given term $\cos 70\sin 48-\cos 48\sin 70$ as $-\sin 22$.
$\therefore $ The value of the given term $\cos 70\sin 48-\cos 48\sin 70$ is $-\sin 22$.

Note:
We should not assume $A=70$ and $B=48$ while solving this problem, as this is the common mistake done by students. We can also solve the given problem by making use of the product to sum formula $2\sin A\cos B=\sin \left( A+B \right)-\sin \left( A-B \right)$, which will also give a similar result. We should not make calculation mistakes while solving this type of problem. Similarly, we can expect problems to find the value of $\sin 48\cos 18-\cos 18\sin 48$.