Question

(sintheta - cosec thetha)
(costheta-sectheta) (tan theta+cot theta) =1​

Answer 1

Answer:

The given is proved,

LHS=RHS

Solution:  

Given Data:

According to the question:

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

Step 1:

To prove,  

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

Proof:

Step 2:

LHS $=\left(\frac{1}{\sin } \theta-\sin \theta\right)\left(\frac{1}{\cos } \theta-\cos \theta\right)\left(\tan \theta+\frac{1}{\tan } \theta\right)$

$=\frac{\frac{1-\sin ^{2} \theta}{\sin ^{2} \theta}\left(1-\cos ^{2} \theta\right)}{\tan \theta}$

Step 3:

   $=\frac{\frac{\cos ^{2} \theta}{\sin \theta}\left(\sin ^{2} \theta\right) \sec ^{2} \theta}{\tan \theta}$

     $=\cos \theta \sin \theta \frac{\frac{1}{\cos ^{2} \theta}}{\frac{\sin \theta}{\cos \theta}}$

$=\cos \theta \sin \theta\left(\frac{1}{\cos ^{2} \theta}\right)\left(\frac{\cos \theta}{\sin \theta}\right)$

Step 4:

$=\cos \theta \sin \theta\left(\cos \frac{\theta}{\sin } \theta \cos ^{2} \theta\right)$

$=\cos \theta \sin \theta\left(\frac{1}{\sin } \theta \cos \theta\right)$

Step 5:

Result:

$=\cos \theta \sin \frac{\theta}{\sin } \theta \cos \theta$

 =1=RHS

∴ Hence proved