Use the sum and difference identities to find the exact value of tan 105?
1 Answer
Apr 14, 2018
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)#