# 1)Find the exact value of the given expression. `sin(pi/3-7pi/4)` 2)Write the given expression as the tangent of an angle: `(tan3x-tan7x) /(1+tan3xtan7x)` 3)Write the given expression as the...

1)Find the exact value of the given expression. `sin(pi/3-7pi/4)`

2)Write the given expression as the tangent of an angle: `(tan3x-tan7x) /(1+tan3xtan7x)`

3)Write the given expression as the tangent of an angle: `(tan2x+tan6x)/(1-tan2xtan6x)`

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The expression for tangent of the sum of two angles a and b is:

`tan(a+b)=(tana+tanb)/(1-tanatanb) `

Therefore:

` (tana+tanb)/(1-tanatanb)=tan(a+b)` (1)

Substitute 2x for a and 6x for b in (1):

`(tan2x+tan6x)/(1-tan2xtan6x)=tan(2x+6x)=tan8x `

**Thus the expression can be written as tan(8x).**

The exact value of `sin(pi/3 - 7*pi/4)` has to be determined.

`sin(pi/3 - 7*pi/4)`

= `sin(pi/3 - pi - 3*pi/4)`

= `sin (-5pi/12 - pi)`

= `-sin(5pi/12 + pi)`

= `sin(5*pi/12)`

= `sin(pi/6 + pi/4)`

= `sin(pi/6)*cos(pi/4) + cos(pi/6)*sin(pi/4)`

= `1/2*(1/sqrt 2) + sqrt3/2*(1/sqrt2)`

= `sqrt 2/4 + sqrt6/4`

= `(sqrt 2+sqrt 6)/4`

2. The expression for the tangent of the difference of two angles a and b is `tan(a-b) = (tan a - tan b)/(1+tan a*tan b)`

`(tan 3x - tan 7x)/(1 + tan 3x*tan 7x) = tan(3x - 7x) = tan -4x = -tan 4x`