Question

sinΦ(1+tanΦ)+cosΦ(1+cotΦ)=secΦ+cosecΦ​

Answer 1

$L.H.S. = \sin \phi(1 +\tan\phi)+\cos \phi(1 +\cot\phi)=sec \phi +cosec\phi$, proved.

Step-by-step explanation:

We have,

$L.H.S. = \sin \phi(1 +\tan\phi)+\cos \phi(1 +\cot\phi)$

$= \sin \phi(1 +\frac{sin\phi}{cos\phi} )+\cos \phi(1 +\frac{\cos\phi}{\sin\phi} )$

$[tex]= \sin \phi(\frac{cos\phi+sin\phi}{cos\phi} )+\cos \phi(\frac{cos\phi+sin\phi}{sin\phi} } )$

$= (\sin \phi + \sin \phi)(\frac{\sin \phi}{\cos \phi} +\frac{\cos \phi}{\sin \phi})$

$= (\sin \phi + \sin \phi)(\tan \phi +\cot \phi )$

$= (\sin \phi + \sin \phi)(\frac{\sin \phi}{\cos \phi}  + \frac{\cos \phi}{\sin \phi} )$

$= (\sin \phi + \sin \phi)(\dfrac{\sin^{2} \phi  + cos^{2} \phi }{\sin \phi \times\ \cos \phi } )$

$= (\sin \phi + \sin \phi)(\dfrac{1}{\sin \phi \times\ \cos \phi } )$

$=\dfrac{1}{\cos \phi } +\dfrac{1}{\sin \phi } =<strong>sec \phi +cosec\phi$

= R.H.S. proved

Answer 2

Answer:

Step-by-step explanation: