Question

I need to count this:
a) 1-tg^2a/1+ctg^2a, when sina =2/3
b) 1-tg^2a/1-ctg^2a, when cosa= -1/3
c) tga * cos^2a, when sina + cosa=5/7
d) tg^2a + 1/sina * 1/cosa + ctg^2a, when tga + ctga =5

Answer 1

a) if sina = 2/3 then tana = 2/√5 and cota = √5/2
So, 1-tan^2 / 1+cot^2 = (1-4/5)/(1+5/4) = 1/5 / 9/4 = 4/45

b) cosa = -1/3 means tana = √8 and cota = 1/√8
so, 1-tan^2 / 1-cot^2 = (1-8)/(1 - 1/8) = -7 / 7/8 = -8

c) If sina + cosa = 5/7 then
(sina + cosa)^2 = 25/49
sin^2a + 2sins cosa + cos^2a = 25/49
1 + 2 sina cosa = 25/49
2 sina cosa = -24/49

Now, tana * cos^2a = sina cosa = -12/49

d) tana + cota = 5

(tana + cota)^2 = tan^2 + cot^2 + 2 = 25
so, tan^2a + cot^2a = 23

tana + cota = 5
sina/cosa + cosa/sina = 5
(sin^2a + cos^2a)/(sina cosa) = 5
1/(sina cosa) = 5

tan^2a + 1/sina * 1/cosa + cot^2 a
= tan^2a + cot^2a + 1/(sina cosa)
= 23 + 5 = 28