Question

prive that (cosecp -sinp) (secp - cosp) (tanp+cotp)=1

Answer 1

Answer: Proved.

Step-by-step explanation:  We are given to prove the following equality involving trigonometric ratios:

$(\csc p-\sin p)(\sec p-\cos p)(\tan p+\cot p)=1.$

We will be using the following identities in the proof:

$(i)~1-\sin^2p=\cos^2p,\\\\(ii)~1-\cos^2p=\sin^2p,\\\\(iii)~\sin^2p+\cos^2p=1.$

We have

$L.H.S.\\\\=(\csc p-\sin p)(\sec p-\cos p)(\tan p+\cot p)\\\\=\left(\dfrac{1}{\sin p}-\sin p\right)\left(\dfrac{1}{\cos p}-\cos p\right)\left(\dfrac{\sin p}{\cos p}+\dfrac{\cos p}{\sin p}\right)\\\\\\=\left(\dfrac{1-\sin^2p}{\sin p}\right)\left(\dfrac{1-\cos^2p}{\cos p}\right)\left(\dfrac{\sin^2p+\cos^2p}{\sin p\cos p}\right)\\\\\\=\dfrac{\cos^2p}{\sin p}\times \dfrac{\sin^2p}{\cos p}\times \dfrac{1}{\sin p\cos p}\\\\=1\\\\=R.H.S.$

Hence proved.