SOLUTION: prove that the given equation is an identity cos((pi/2)-x)=sinx

Question 1163196: prove that the given equation is an identity
cos((pi/2)-x)=sinx
Found 2 solutions by solver91311, Theo:
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Hint: Use the formula for the cosine of the difference of two angles. You will end up with 0 + sin(x).

John

My calculator said it, I believe it, that settles it

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
this can be solved geometrically, but you can also solve it using trig identities.

the trig identity you can use is:

cos(a - b) = cos(a) * cos(b) + sin(a) * sin(b)

the equation you want to prove is true is:

cos(pi/2) - x) = sin(x)

let a = pi/2 and b = x

the identity of:

cos(a - b) = cos(a) * cos(b) + sin(a) * sin(b) becomes:

cos(pi/2 - x) = cos(pi/2) * cos(x) + sin(pi/2) * sin(x)

cos(pi/2) = 0
sin(pi/2) = 1

cos(pi/2 - x) = cos(pi/2) * cos(x) + sin(pi/2) * sin(x) becomes:
cos(pi/2 - x) = 0 * cos(x) + 1 * sin(x)
simplify to get:
cos(pi/2 - x) = sin(x)

there's your proof.

a nice list of trigonometric identities can be found at:
https://www.purplemath.com/modules/idents.htm

the identity you are looking for will be under the title of angle sum and difference identities.

you can also solve this geometrically as follows:

draw a right triangle ABC with the right angle at C.
since the sum of the angles of a triangle = 180, then:
angle A + angle B + angle C = 180
since angle C = 90 degrees, then:
angle A + angle B + 90 degrees = 180 degrees
subtract 90 from both sides of that equation to get:
angle A + angle B = 90 degrees.
solve for angle B to get:
angle B = 90 - angle A.
in the triangle ABC, the side opposite angle A is side a, the side opposite angle B is side b, the side opposite angle C is side c.
since C is the 90 degree angle, then the hypotenuse of the triangle is side c.
cos(angle B) = adjacent / hypotenuse = side a divided by side c
sin(angle A) = opposite / hypotenuse = side a divided by side c
this makes cos(angle B) = sin(angle A)
since angle B = 90 - angle A, this makes cos(90 - angle A) = sin(angle A)
let angle A = x, then this equation becomes:
cos(90 - x) = sin(x)
translate 90 degrees to radians to get:
90 degrees * pi / 180 = pi/2 radians.
replace 90 degrees with pi/2 radians to get cos(pi/2 - x) = sin(x)
this also proves the equation is an identity.

here's a diagram that helps you to see what's happening with the geometric proof.

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