Question

Which of the following are identities? Check all that apply.

a) tan x - tan y = sin(x-y)/cosxcosy
b) 1 - tanxtany = sin(x+y)/sinxsiny
c) tan(x-pie/4) = tanx-1
d) cos(x+pie/6) = -sin(x-pie/3)

Answer 1

a) tan x - tan y = sin(x-y)/cos(x)cos(y)
b) 1 - tan(x)tan(y) = sin(x+y)/sin(x)y
c) tan(x-pi/4) = tanx-1
d) cos(x+pi/6) = -sin(x-pi/3)

Only a) is an identity.

Step-by-step explanation:

a) tan x - tan y = sin(x-y)/cos(x)cos(y)

This is an identity. To prove it, we can expand the right side of the equation using trigonometric identities:
sin(x-y)/cos(x)cos(y) = (sinx*cosy - cosx*siny)/(cosx*cosy) = sinx/cosx - siny/cosy = tanx - tany

b) 1 - tan(x)tan(y) = sin(x+y)/sin(x)y

This is not an identity. To disprove it, we can choose specific values of x and y and see if the equation holds true. For example, if we set x=0 and y=90, the left side of the equation becomes 1 - 0 = 1, while the right side becomes sin(0+90)/sin(0)sin(90) = sin(90)/0*sin(90) = undefined.

c) tan(x-pi/4) = tanx-1

This is not an identity. To disprove it, we can again choose specific values of x and see if the equation holds true. For example, if we set x=0, the left side of the equation becomes tan(-pi/4) = tan(-45 degrees) = -1, while the right side becomes tan(0) - 1 = 0 - 1 = -1.

d) cos(x+pi/6) = -sin(x-pi/3)

This is not an identity. To disprove it, we can again choose specific values of x and see if the equation holds true. For example, if we set x=0, the left side of the equation becomes cos(pi/6) = cos(30 degrees) = sqrt(3)/2, while the right side becomes -sin(-pi/3) = -sin(60 degrees) = -sqrt(3)/2. Since sqrt(3)/2 is not equal to -sqrt(3)/2, the equation is not an identity.

Answer 2

The correct answer option is a) tan x - tan y = sin(x-y)/cosxcosy.

Step-by-step explanation:

The option a) tan x - tan y = sin(x-y)/cosxcosy is an identity. Let us prove it:

The right side is equal to:

`(sin(x)cos(y)-cos(x)sin(y))/(cos(x)cos(y))`

`=(sin(x))/(cos(x))-(sin(y))/(cos(y))`

`=tan(x)-tan(y)`

which is equal to the left side.

Hence, tan x - tan y = sin(x-y)/cosxcosy is an identity.

While rest of the options do not equal to each of their sides so they are not identities.