Best solutions to - "How to simplify the expression sin⁡t1−cos⁡t+1−cos⁡tsin⁡t?"

disciple11k

Answered question

2023-01-15

How to simplify the expression

\frac{\sin t}{1 - \cos t} + \frac{1 - \cos t}{\sin t}

Digngoorpr43

Beginner2023-01-16Added 14 answers

= \frac{(\sin t) (\sin t)}{\sin t (1 - \cos t)} + \frac{(1 - \cos t) (1 - \cos t)}{\sin t (1 - \cos t)}

= \frac{\sin^{2} t + 1 - 2 \cos t + \cos^{2} t}{\sin t (1 - \cos t)}

Employ the identity

\sin^{2} θ + \cos^{2} θ = 1

= \frac{1 + 1 - 2 \cos t}{\sin t (1 - \cos t)}

= \frac{2 - 2 \cos t}{\sin t (1 - \cos t)}

Factor:

= \frac{2 (1 - \cos t)}{\sin t (1 - \cos t)}

= \frac{2}{\sin t}

Use

\csc θ = \frac{1}{\sin θ}

= 2 \csc t

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