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Which of the following are identities? Check all that apply. a) tan(x-pi)=tanx b) sin(x+y)+sin(x-y)=2cosxsiny c) cos(x+y)+cos(x-y)=2cosxcosy d) cos(x+y)-cos(x-y)=2cosxcosy

Respuesta :

The trigonometric identities that apply are:

Option A and Option C

- tan(x-pi)=tanx

- cos(x+y)+cos(x-y)=2cosxcosy

Given,

- tan(x-pi)=tanx

- sin(x+y)+sin(x-y)=2cosxsiny

- cos(x+y)+cos(x-y)=2cosxcosy

- cos(x+y)-cos(x-y)=2cosxcosy

We have to check which identities are correct.

1.

tan(x-pi)=tanx

tan(x-pi) = tanx.      

[tan(-x) = -tanx]

tan(-(pi-x)) = tanx

-tan(pi-x) = tanx.      

[tan(pi - x) lies in the 2nd quadrant and it is negative]

-(-tanx) = tanx

tanx = tanx

This is an identity.

2.

sin(x+y)+sin(x-y)=2cosxsiny                

sin(x+y) + sin(x-y) = 2cosxsiny.        

[sin(A+B) = sinAcosB + cosAsinB]

sinxcosy + cosxsiny + sinxcosy - cosxsiny = 2cosxsiny

2sinxcosy ≠ 2cosxsiny

This is not an identity.

3.

cos(x+y)+cos(x-y)=2cosxcosy.      

[cos(A+B) = cosAcosB - sinAsinB]

[cos(A-B) = cosAcosB + sinAsinB]

cosxcosy - sinxsiny + cosxcosy + sinxsiny = 2cosxcosy

2cosxcosy = 2cosxcosy

This is an identity.

4.

cos(x+y)-cos(x-y)=2cosxcosy

cosxcosy - sinxsiny - (cosxcosy + sinxsiny) = 2cosxcosy

cosxcosy - sinxsiny - cosxcosy - sinxsiny = 2cosxcosy

-2sinxsiny ≠ 2cosxcosy

This is not an identity.

Thus,

Option A and Option C are true.

- tan(x-pi)=tanx

- cos(x+y)+cos(x-y)=2cosxcosy

Learn more about trigonometric identities here:

https://brainly.com/question/12526043

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