Question

there comes this step while solving a question of Calculus : sin(x+deltax)cosx-cos(x+deltax)sinx which when simplified becomes: sin(x+deltax-x). Kindly explain this step of simplification to me! I am very confused as I cannot understand this. whether a formula applies or you open the brackets(but incase you open brackets how can you multiply sin and cosine with x+deltax?)

Answer 1

Hey there! :)
So the formula that was applied was the Sine Angle Difference Formula:\[\large\rm \sin(A-B)=\sin A \cos B- \sin B \cos A\]It's being applied in reverse.

Answer 2

Oh color might make it a little more clear, sec.

Answer 3

\[\large\rm \sin(\color{royalblue}{A}-\color{orangered}{B})=\sin \color{royalblue}{A} \cos \color{orangered}{B}- \sin \color{orangered}{B} \cos \color{royalblue}{A}\]

Answer 4

So for our problem,\[\large\rm \sin(\color{royalblue}{x+\Delta x}-\color{orangered}{x})=\sin (\color{royalblue}{x+\Delta x}) \cos \color{orangered}{x}- \sin \color{orangered}{x} \cos (\color{royalblue}{x+\Delta x})\]

Answer 5

Thank you so much :)