Verify the following identity:
(tan(x) + 1)/sin(x) - sec(x) = csc(x)
Put (tan(x) + 1)/sin(x) - sec(x) over the common denominator sin(x): (tan(x) + 1)/sin(x) - sec(x) = (1 - sec(x) sin(x) + tan(x))/sin(x):
(1 - sec(x) sin(x) + tan(x))/sin(x) = ^?csc(x)
Multiply both sides by sin(x):
1 - sec(x) sin(x) + tan(x) = ^?csc(x) sin(x)
Write cosecant as 1/sine, secant as 1/cosine and tangent as sine/cosine:
1 - 1/cos(x) sin(x) + sin(x)/cos(x) = ^?1/sin(x) sin(x)
1 - (1/cos(x)) sin(x) + (sin(x)/cos(x)) = 1:
1 = ^?(1/sin(x)) sin(x)
(1/sin(x)) sin(x) = 1:
1 = ^?1
The left hand side and right hand side are identical:
(identity has been verified)
(1 + tan x ) / sin x - sec x = csc x
(1 + tan x) / sin x - 1 / cos x
(1 + sin x / cos x) / sin x - 1 / cos x
1/sin x + sinx / [ cos x sin x] - 1/cos x
1 / sin x + sin x / [ cos x sin x ] - sin x / [ sinx cos x]
1 / sin x + sin x [ sin x cos x] - sin x / [ sin x cos x ]
1 / sin x = csc x