1,

a, `tan(100)+1/(tan(80))−tan(10)−1/(tan(20))` = -8,4188

b, `tan(1234)+1/(tan(4321))` = 56,8022

2,

a, `3*tgx-ctgx = 0`

Felt: `x ne (k*pi)/2`

`3*tgx-1/(tgx)=0`

tg x = a helyettesítéssel

`3a-1/a=0` (a kezdeti feltétel miatt lehet szorozni a-val)

`3a^2-1=0`

`3a^2=1`

`a^2=1/3`

`a_1` = `root()(3)/3` = tgx

`x_1` = `pi/6+k*pi`

`a_2` = `-root()(3)/3` = tgx

`x_2` = `-pi/6+k*pi`

Megoldás:

x = `pm pi/6+k*pi`, ahol `k in ZZ`.


b,

`tgx+ctgx=2`

Felt: `x ne (k*pi)/2`

`tgx+1/(tgx)=2`

tgx = a helyettesítéssel

`a+1/a=2`

`a^2-2a+1=0`

`(a-1)^2=0`

a-1 = 0

a = 1 = tgx `rightarrow` x = `pi/4+k*pi`, ahol `k in ZZ`

c,

`tg^2 x-6*tgx+5=0`

Felt: `x ne pi/4+k*pi`

tgx = a

`a^2-6a+5=0`

(a-1)(a-5)=0

szorzat akkor nulla, ha valamelyik tényezője nulla.

`a_1` = 1 = tgx

`x_1` = `pi/4+k*pi`

`a_2` = 5 = tgx

`x_2` = `1,37 + k*pi`, ahol `k in ZZ` (78,69° fokban)