Question

To derive the tangent formula, the following steps are given:
1. $\tan \left(A+B\right)=\frac{\frac{\sin A\cos B}{\cos A\cos B}+\frac{\cos A\sin B}{\cos A\cos B}}{\frac{\cos A\cos B}{\sin A\sin B}+\frac{\sin A\sin B}{\cos A\cos B}}$
2. $\tan \left(A+B\right)=\frac{\sin \left(A+B\right)}{\cos \left(A+B\right)}$
3. $\tan \left(A+B\right)=\frac{\sin A\cos B+\cos A\sin B}{\cos A\cos B-\sin A\sin B}$
4.$\tan \left(A+B\right)=\frac{\tan A+\tan B}{1-\tan A\tan B}$
Their correct and proper sequential form to derive the formula is:

• A. 2, 4, 3, 1
• B. 1, 2, 3, 4
• C. 1, 4, 2, 3
• D. 2, 3, 1, 4

Answer 1

Answer: (D)

Tangent formula is derived as follows
$\tan \left(A+B\right)=\frac{\sin \left(A+B\right)}{\cos \left(A+B\right)}$
$=\frac{\sin A\cos B+\cos A\sin B}{\cos A\cos B-\sin A\sin B}$
$=\frac{\frac{\sin A\cos B}{\cos A\cos B}+\frac{\cos A\sin B}{\cos A\cos B}}{\frac{\cos A\cos B}{\cos A\cos B}-\frac{\sin A\sin B}{\cos A\cos B}}=\frac{\tan A+\tan B}{1-\tan A\tan B}$
Correct and proper sequential form to derive the formula is 2, 3, 1, 4.