Find the exact value of the expression by using appropriate identities. Do not use a calculator. (tan 77 + tan 43)/(1 - tan 77.tan 43)

Question

Find the exact value of the expression by using appropriate identities. Do not use a calculator.
`(tan 77 + tan 43)/(1 - tan 77.tan 43)`

Answer 1

Explanation:

Remember from Trig. identity:

Tan(a + b) = (tan a + tan b)/1 - tana × tan b

From the above,

(tan 77 + tan 43)/1 - tan 77 × tan 43

Comparing Both equations on the LHS,

a = 77

b = 43

Therefore, on the right hand side;

Tan (a + b) = tan (77 + 43)

= tan 120

Tan θ = sin θ/cos θ

Sin 120 = (root3)/2

Cos 120 = -1/2

Therefore, tan 120 = (root3)/2 × -2/1

= -(root3)

Answer 2

`-√(3)`

Explanation:

From Trigonometric Identity

`Tan (A+B)=(tan A + tan B)/(I-tan A\cdot tan B)`

Therefore comparing
`(tan 77 + tan 43)/(I-tan 77\cdot tan 43)` with the above:

A=77, B=43

`(tan 77 + tan 43)/(I-tan 77\cdot tan 43)=Tan (77+43)`

=Tan 120

Tan is Negative in the Second Quadrant

Tan 120 =
`-√(3)`