Which of the following counterexamples proves that sinxtanx=cosx is not a trigonometric identity? Select all that apply.

Question

-2π
-3π
-3π/4
-π/4

Answer 1

For x = - 2 π

sin ( - 2 π ) = 0 , cos ( - 2 π ) = 1 , tan ( - 2 π ) = 0

sin ( - 2 π ) ∙ tan ( - 2 π ) = 0 ∙ 0 = 0 ≠ cos ( - 2 π )

0 ≠ 1

For x = - 3 π

sin ( - 3 π ) = 0 , cos ( - 3 π ) = - 1 , tan ( - 3 π ) = 0

sin ( - 3 π ) ∙ tan ( - 3 π ) = 0 ∙ 0 = 0 ≠ cos ( - 3 π )

0 ≠ - 1

For x = - 3 π / 4

sin ( - 3 π / 4 ) = -√ 2 / 2 , cos ( - 3 π / 4 ) = - √ 2 / 2 , tan ( - 3 π / 4 ) = 1

sin ( - 3 π / 4 ) ∙ tan ( - 3 π / 4 ) = -√ 2 / 2 ∙ 1 = - √ 2 / 2 = cos ( - 3 π / 4 )

- √ 2 / 2 = - √ 2 / 2

For x = - π / 4

sin ( - π / 4 ) = -√ 2 / 2 , cos ( - π / 4 ) = √ 2 / 2 , tan ( - π / 4 ) = - 1

sin ( - π / 4 ) ∙ tan ( - π / 4 ) = -√ 2 / 2 ∙ ( - 1 ) = √ 2 / 2 = cos ( - π / 4 )

√ 2 / 2 = √ 2 / 2

So x = - 2 π and x = - 3 π proves that

sin x ∙ tan x = cos x is not a trigonometric identity,

Answer 2

Seems like Reiny already helped you with this, but here goes again.
If it is an identity, then all of the values suggested should work.
So, does
sin(-2π)*tan(-2π) = cos(-2π) ?
0*0 = 1 ?
Nope. So, using -2π proves it is not an identity.

Now you try the others. If you get stuck, come on back and show what you got.

Answer 3

Also you can do this:

sin x ∙ tan x = cos x

sin x ∙ sin x / cos = cos x

sin² x / cos = cos x

Multiply both sides by cos x

sin² x = cos² x

For x = - 2 π

sin ( - 2 π ) = 0 , cos ( - 2 π ) = 1

sin² ( - 2 π ) = 0 ≠ cos² ( - 2 π )

0 ≠ 1²

0 ≠ 1

For x = - 3 π

sin ( - 3 π ) = 0 , cos ( - 3 π ) = - 1 ,

sin² ( - 3 π ) = 0 ≠ cos² ( - 3 π )

0 ≠ ( - 1 )²

0 ≠ 1

For x = - 3 π / 4

sin ( - 3 π / 4 ) = -√ 2 / 2 , cos ( - 3 π / 4 ) = - √ 2 / 2

sin² ( - 3 π / 4 ) = ( -√ 2 / 2 )² = 2 / 4 = 1 / 2 =

cos² ( - 3 π / 4 ) = ( - √ 2 / 2 )² = 2 / 4 = 1 / 2

1 / 2 = 1 / 2

For x = - π / 4

sin ( - π / 4 ) = -√ 2 / 2 , cos ( - π / 4 ) = √ 2 / 2

sin² ( - π / 4 ) = ( -√ 2 / 2 )² = 2 / 4 = 1 / 2 =

cos² ( - π / 4 ) = ( - √ 2 / 2 )² = 2 / 4 = 1 / 2

1 / 2 = 1 / 2

So x = - 2 π and x = - 3 π proves that

sin x ∙ tan x = cos x is not a trigonometric identity.

Answer 4

Or, having arrived at
sin² x = cos² x
then
tan² x = 1
which is only true for odd multiples of π/4
so, for -2π and -3π it won't work