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Question

sinx+cosx=1/sinx

Answer 1

sinx+cosx=1/sinx
Multiply each side by sinx

sinx(sinx+cosx)=1

sin^2(x) +sinxcosx =1

sinxcosx = 1- sin^2(x)

Using pythagorean identity:1- sin^2(x) = cos^2(x)

sinxcosx = cos^2(x)

sinxcosx - cos^2(x) = 0

cosx (sinx - cos x) =0

cosx = 0      or      sinx - cos x = 0 ⇒ tan x =1

cosx = 0 

General solution is  x = (2n + 1)π / 2

tan x =1

tan x = tan π / 4

General solution is  x = nπ + π / 4.