Question

Students were asked to prove the identity (sec x)(csc x) = cot x + tan x. ​

Answer 1

Let's prove that (sec x)(csc x) is equal to cot x + tan x

$\Longrightarrow \sf (sec(x) )(csc(x))$

$\Longrightarrow \sf \dfrac{1}{\cos \left(x\right)\sin \left(x\right)}$

$\Longrightarrow \sf \dfrac{\cos ^2\left(x\right)+\sin ^2\left(x\right)}{\cos \left(x\right)\sin \left(x\right)}$

$\Longrightarrow \sf \dfrac{\cos ^2\left(x\right)}{\cos \left(x\right)\sin \left(x\right)} + \dfrac{\sin ^2\left(x\right)}{\cos \left(x\right)\sin \left(x\right)}$

$\Longrightarrow \sf \dfrac{\cos\left(x\right)}{\sin \left(x\right)} + \dfrac{\sin \left(x\right)}{\cos \left(x\right)}$

$\Longrightarrow \sf cot(x) + tan(x)$

Hence student A did correctly prove the identity properly.

Also Looking at student B's work, he verified the identity properly.

So, Both are correct in their own way.

Part B

Identities used:

$\rightarrow \sf sin^2 (x) + cos^2 (x) = 1$       (appeared in step 3)

$\sf \rightarrow \dfrac{cos(x) }{sin(x) } = cot(x)$               (appeared in step 6)

$\rightarrow \sf \dfrac{sin(x )}{cos(x) } = tan(x)$               (appeared in step 6)