1. ## prove it

sin theta(1 + tan theta) + cos theta(1 + cot theta) = sec theta + cosec theta

plzz complete it as i am new to this chapter

2. Originally Posted by saha.subham
sin theta(1 + tan theta) + cos theta(1 + cot theta) = sec theta + cosec theta

plzz complete it as i am new to this chapter

It is just for easier input , i put $\displaystyle \theta = x$ so what we are going to prove is :

$\displaystyle \sin{x} ( 1 + \tan{x} ) + \cos{x} ( 1+ \cot{x} ) = \sec{x} + \csc{x}$

Consider

$\displaystyle \tan{x} = \frac{\sin{x}}{\cos{x}}$ and

$\displaystyle \cot{x} = \frac{\cos{x}}{\sin{x}}$

So

$\displaystyle 1 + \tan{x} = 1 + \frac{\sin{x}}{\cos{x}} = \frac{ \sin{x} + \cos{x} }{ \cos{x}}$

$\displaystyle 1 + \cot{x} = 1 + \frac{\cos{x}}{\sin{x}} = \frac{ \sin{x} + \cos{x} }{ \sin{x}}$

and we have

$\displaystyle \sin{x} ( 1 + \tan{x} ) + \cos{x} ( 1+ \cot{x} ) =$

$\displaystyle \sin{x}~\frac{ \sin{x} + \cos{x} }{ \cos{x}} + \cos{x} ~ \frac{ \sin{x} + \cos{x} }{ \sin{x}}$

Look at the common factor : $\displaystyle \sin{x} + \cos{x}$ get them out and we obtain :

$\displaystyle (\sin{x} + \cos{x}) \left(\frac{ \sin{x}}{\cos{x}} + \frac{ \cos{x}}{\sin{x}} \right)$

$\displaystyle = (\sin{x} + \cos{x}) ~ \frac{ \sin^2{x} + \cos^2{x} }{\sin{x}\cos{x}}$

I think you can finish it starting from here .

3. ## another way

sinx(1+tanx)+cosx(1+cotx)
=sinx(1+tanx)+cosx[(1+tanx)/tanx]............................ [cotx=1/tanx]
=(1+tanx)[sinx+(cosx/tanx)]
=(1+tanx)[(sinx)^2+(cosx)^2]/sinx............................. [tanx=sinx/cosx]
=(1+tanx)/sinx
=((1/sinx)+(tanx/sinx))
=secx+cosecx

4. Did someone call me?