Question

To prove :- sin (x+30) = cos x + sin(x-30)​

Answer 1

Step-by-step explanation:

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant.

Answer 2

to prove the following is true

( sin x)( sin [ x +30°]) + (cos x)( sin[ x +120°]) = √3/2

replace sin [ x + 120°] with cos [ x + 30°]

since sin [x + 30° + 90°[= cos [ x + 30°]

gives us this equation to prove

(sin x)( sin [ x +30°]) + ( cos x)( cos [x + 30°]) = √3/2

replace x + 30° with -[ x + 30°]

cos [ x + 30°] = cos -[ x + 30°], cosine is an even function

sin [ x + 30°] = - sin -[x + 30°], sine is an odd function

Making the above substitutions, this is the equation we must prove

(cos)( cos -[x +30°]) - ( sin x)( sin -[x + 30]) = √3/2

Note

(cos)( cos -[x +30°]) - ( sin x)( sin -[x + 30]) = cos ( x-[x+30] = cos -30°

cos -30° = cos 30°

and

cos 30° = √3/2

so we have proved the equation

( sin x)( sin [ x +30°]) + (cos x)( sin[ x +120°]) = √3/2