Use the sum and difference identities to find the exact value of tan 105?

1 Answer
Apr 14, 2018

#-(2+sqrt3)#

Explanation:

#"using the "color(blue)"trigonometric identity"#

#•color(white)(x)tan(A+B)=(tanA+tanB)/(1-tanAtanB)#

#"note that " 105^@=(60+45)^@#

#rArrtan105^@=tan(60+45)^@#

#=(tan60^@+tan45^@)/(1-tan60^@tan45^@)#

#=(sqrt3+1)/(1-sqrt3)#

#color(blue)"rationalise the denominator"#

#"multiply numerator denominator by the "color(blue)"conjugate"#
#"of the denominator"#

#"the conjugate of "1-sqrt3" is "1color(red)(+)sqrt3#

#rArr((1+sqrt3)(1+sqrt3))/((1-sqrt3)(1+sqrt3))#

#=(1+2sqrt3+3)/(1-3)=(4+2sqrt3)/(-2)#

#color(white)(xxxxxxxxxxxx)=-2-sqrt3=-(2+sqrt3)#