Question

prove that √1+cos theta/√1–cos theta = cot * theta /2​

Answer 1

Answer:

Step-by-step explanation:

\begin{gathered}to \: prove \\ \\ \frac{ \sqrt{1 + \cos\alpha } }{ \sqrt{1 - \cos \alpha } } = cosec \alpha + \cot \alpha \\ \\ proof \\ \\ lhs \\ \\ = \frac{ \sqrt{1 + \cos \alpha } }{ \sqrt{1 - cos \alpha } } \\ \\ = \frac{ \sqrt{(1 + cos \alpha )(1 + cos \alpha )} }{ \sqrt{(1 - cos \alpha )(1 + cos \alpha )} } \\ \\ = \frac{ \sqrt{ {(1 + cos \alpha )}^{2} } }{ \sqrt{ {1}^{2} - {cos}^{2} \alpha } } \\ \\ = \frac{ \sqrt{( {1 + cos \alpha )}^{2} } }{ \sqrt{ {sin}^{2} \alpha } } \\ \\ = \frac{1 + cos \alpha }{ \sin\alpha } \\ \\ = \frac{1}{sin \alpha } + \frac{cos \alpha }{sin \alpha } \\ \\ = cosec \alpha + cot \alpha \\ \\ = rhs \\ \\ hence \: proved\end{gathered}

toprove

1−cosα

1+cosα

=cosecα+cotα

proof

lhs

=

1−cosα

1+cosα

=

(1−cosα)(1+cosα)

(1+cosα)(1+cosα)

=

1

2

−cos

2

α

(1+cosα)

2

=

sin

2

α

(1+cosα)

2

=

sinα

1+cosα

=

sinα

1

+

sinα

cosα

=cosecα+cotα

=rhs

henceproved

=================================

Have taken theta =α

FORMULAS USED:

1) (a+b)(a-b) =a²-b²

2)cos²A+sin²A=1

3)1/sinA=cosecA

4)cosA/sinA=cotA