Question

Prove the identity. (tanx/1−cosx) = (cscx)(1+secx)

Answer 1

rewrite the RHS in terms of sin and cosine..

Answer 2

\[\frac{ 1 }{ sinx}(1+\frac{ 1 }{\cos x})\]

Answer 3

\[\frac{ 1 }{sinx}(\frac{ \cos x+1 }{ \cos x })\]

Answer 4

does this make sense?

Answer 5

We can easily convert LHS into RHS...

Answer 6

Let us start from LHS Rationalize it by multiplying it with (1 + cosx)..

\[\frac{\tan(x)}{1 - \cos(x)} \times \frac{1 + \cos(x)}{1 + \cos(x)} = \frac{\tan(x)(1 + \cos(x))}{1 - \cos^2(x)}\]

Answer 7

\[\implies \frac{\tan(x)(1 + \cos(x))}{\sin^2(x)} \implies \frac{\sin(x)(1 + \cos(x))}{\cos(x) \cdot \sin^2(x)} \implies \frac{cosec(x)(1 + \cos(x))}{\cos(x)}\]

Answer 8

Using :

\[\frac{1}{\sin(x)} = cosec(x)\]
\[\frac{1}{\cos(x)} = \sec(x)\]

Answer 9

\[\implies {cosec(x)} \times (\sec(x) + \sec(x) \cdot \cos(x)) \implies "Hence \; Proved.."\]

Answer 10

\[\sec(x) \cdot \cos(x) = 1\]