Verify the identity tanx coty tanx coty tany cotx Find a

Verify the identity tan(x) + cot(y) = tan(x) cot(y) = tan(y) + cot(x) Find all the solutions for sec x + tan x = 1

Solution

11) (tan x + cot y)/(tan x*cot y) = tan y + cot x

LHS = (tan x + cot y)/(tan x*cot y) = 1/cot y   + 1/tan x = tan y + cot x = RHS

12) tan(x)+sec(x)=1
sin(x) / cos(x) + 1/cos(x)=1
(sin(x)+1)/cos(x)=1
sin(x) + 1 = cos(x)
(sin(x)+1)^2 = (cos (x))^2
sin^2 (x) + 2sin (x) + 1 = cos^2 (x)
= 1-sin^2 (x)
sin ^2 (x) +sin ^2(x) +2sin(x) +1-1=0
2sin^2(x) + 2sin(x)=o
2sin(x) ( sin(x) + 1)=o
2sin x = 0 and sin(x)+1=0
sin(x)= 0 and sin(x) = -1
x = pi and 2pi x= 3pi/2 and pi/2