Question

Find the value of following expression (a)tan 15° + tan 75° (a)tan 15°+tan 195°

Answer 1

I'll do part (a) to get you started.

`\tan(A-B) = (\tan(A)-\tan(B))/(1+\tan(A)\tan(B))\\\\\\\tan(45-30) = (\tan(45)-\tan(30))/(1+\tan(45)\tan(30))\\\\\\\tan(15) = (1-(√(3))/(3))/(1+1*(√(3))/(3))\\\\\\\tan(15) = (1-(√(3))/(3))/(1+(√(3))/(3))\\\\\\`

`\tan(15) = ((3)/(3)-(√(3))/(3))/((3)/(3)+(√(3))/(3))\\\\\\\tan(15) = ((3-√(3))/(3))/((3+√(3))/(3))\\\\\\\tan(15) = (3-√(3))/(3)*(3)/(3+√(3))\\\\\\\tan(15) = (3-√(3))/(3+√(3))\\\\\\`

`\tan(15) = ((3-√(3))(3-√(3)))/((3+√(3))(3-√(3)))\\\\\\\tan(15) = ((3)^2 - 2*3*√(3)+(√(3))^2)/((3)^2 - (√(3))^2)\\\\\\\tan(15) = (9 - 6*√(3)+3)/(9 - 3)\\\\\\\tan(15) = (12 - 6*√(3))/(6)\\\\\\\tan(15) = (6(2 - √(3)))/(6)\\\\\\\tan(15) = 2 - √(3)\\\\\\`

Follow similar steps to determine that
`\tan(75) = 2 + √(3)`

Therefore,

`\tan(15)+\tan(75) = (2 - √(3))+(2 + √(3)) = 4`

The final answer to part (a) is 4