Question

Use the definitions sinθ=y/r, cosθ=x/r and/or tanθ=y/x to prove that cotθsecθ=cscθ.

Answer 1

Use the Pythagorean identity sin² θ + cos² θ = 1 to show that 1 + cot² θ = csc ² θ.>>

GIVEN

sin² θ + cos² θ = 1

Divide both sides by sin^2 θ

1 + cos² θ/sin² θ = 1/sin² θ

and since

cos² θ/sin² θ = cot² θ

and

1/sin² θ = csc² θ

then the above becomes

1 + cot² θ = csc ² θ

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Use definitions sinθ = y/r, cosθ = x/r and/or tanθ = y/x to prove that cotθsecθ = cscθ

NOTE that

cotθ = cosθ/sinθ = (x/r)/(y/r) = x/y

and

secθ = 1/cosθ = 1/(x/r) = r/x

and

cscθ = 1/sinθ = 1/(y/r) = r/y

Then

(x/y)( r/x) = r/y

Simplifying the left hand side of the equation,

r/y = r/y

Therefore, since cscθ = r/y, then

cotθsecθ = cscθ.