Question

show that ,cot theta .cos theta +sin theta = cosec theta

Answer 1

cot @*cos@ + sin @ = cos^2 @ / sin @ + sin @
.....(writing cot @ = cos @/sin @)

Now,
cos ^2@/sin @ + sin @ = cos^@ + sin^@ / sin @
= 1/sin @ = cosec @ = RHS

Hope this helps you !

Answer 2

We have to prove that :

$\cot \alpha . \cos\alpha + \sin\alpha = \csc\alpha$

On taking LHS :

$\cot \alpha . \cos \alpha + \sin \alpha \\ \\ = > \frac{ \cos \alpha }{ \sin \alpha } . \cos\alpha + \sin \alpha \\ \\ = > \frac{ { \cos }^{2} \alpha }{ \sin\alpha } + \sin \alpha \\ \\ = > \frac{ { \cos }^{2} \alpha + { \sin }^{2} \alpha }{ \sin \alpha } \\ \\ = > \frac{1}{ \sin \alpha } \\ \\ As \: we \: know \: that: \: { \sin }^{2} \alpha + { \cos }^{2} \alpha = 1 \\ \\ = > \csc\alpha = RHS \\ \\ HENCE \: PROVED$