cos t cos t + sin t − 1 − sin t cos t + sin t − 1 + 1 cos t + sin t − 1 = = cos t − sin t + 1 cos t + sin t − 1 = cos t − sin t + 1 cos t + sin t − 1 ⋅ cos t + sin t + 1 cos t + sin t + 1 = = ( cos t + 1 ) 2 − sin 2 t ( cos t + sin t ) 2 − 1 = cos 2 t + ( 1 − sin 2 t ) + 2 cos t sin 2 t = = 2 cos t + 2 cos 2 t 2 sin t cos t = 1 sin t + cos t sin t = csc t + cot t \begin{array}{l}
\frac{\cos t}{\cos t + \sin t - 1} - \frac{\sin t}{\cos t + \sin t - 1} + \frac{1}{\cos t + \sin t - 1} = \\
= \frac{\cos t - \sin t + 1}{\cos t + \sin t - 1} = \frac{\cos t - \sin t + 1}{\cos t + \sin t - 1} \cdot \frac{\cos t + \sin t + 1}{\cos t + \sin t + 1} = \\
= \frac{(\cos t + 1)^2 - \sin^2 t}{(\cos t + \sin t)^2 - 1} = \frac{\cos^2 t + (1 - \sin^2 t) + 2 \cos t}{\sin 2 t} = \\
= \frac{2 \cos t + 2 \cos^2 t}{2 \sin t \cos t} = \frac{1}{\sin t} + \frac{\cos t}{\sin t} = \csc t + \cot t \\
\end{array} c o s t + s i n t − 1 c o s t − c o s t + s i n t − 1 s i n t + c o s t + s i n t − 1 1 = = c o s t + s i n t − 1 c o s t − s i n t + 1 = c o s t + s i n t − 1 c o s t − s i n t + 1 ⋅ c o s t + s i n t + 1 c o s t + s i n t + 1 = = ( c o s t + s i n t ) 2 − 1 ( c o s t + 1 ) 2 − s i n 2 t = s i n 2 t c o s 2 t + ( 1 − s i n 2 t ) + 2 c o s t = = 2 s i n t c o s t 2 c o s t + 2 c o s 2 t = s i n t 1 + s i n t c o s t = csc t + cot t If cos t + sin t + 1 = 0 then 2 sin ( t + π 4 ) = − 1 ⇒ t = 5 π / 4 or t = − π / 2 \text{If } \cos t + \sin t + 1 = 0 \text{ then } \sqrt{2} \sin \left(t + \frac{\pi}{4}\right) = -1 \Rightarrow t = 5\pi/4 \text{ or } t = -\pi/2 If cos t + sin t + 1 = 0 then 2 sin ( t + 4 π ) = − 1 ⇒ t = 5 π /4 or t = − π /2 cos t − sin t + 1 cos t + sin t − 1 = − 2 sin t − 2 = sin t = [ t = − π / 2 ] = − 1 csc t + cot t = 1 sin t + cos t sin t = 1 + cos t sin t = − 1 \begin{array}{l}
\frac{\cos t - \sin t + 1}{\cos t + \sin t - 1} = \frac{-2 \sin t}{-2} = \sin t = [t = -\pi/2] = -1 \\
\csc t + \cot t = \frac{1}{\sin t} + \frac{\cos t}{\sin t} = \frac{1 + \cos t}{\sin t} = -1 \\
\end{array} c o s t + s i n t − 1 c o s t − s i n t + 1 = − 2 − 2 s i n t = sin t = [ t = − π /2 ] = − 1 csc t + cot t = s i n t 1 + s i n t c o s t = s i n t 1 + c o s t = − 1
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