Question

prove that sinx(1+tanx) +cosx(1+cotx) =secx+cosecx

Answer 1

Answer:      

We have to prove that, sinx(1+tanx) +cosx(1+cotx) =secx+cosecx

L.H.S. =  $sinx (1+tanx) +cosx (1+cotx)$

$= sinx(1+\frac{sin x}{cos x}) + cosx (1 +\frac{cosx}{sin x} )$

$= sin x(\frac{cos x+ sin x}{cos x} ) + cosx(\frac{sinx+cosx}{sinx} )$

$= tanx(cosx+sinx)+cotx(sinx+cosx)$

$= (tanx+cotx)(sinx+cosx)$

$=(tanx+cotx)(\frac{1}{secx} + \frac{1}{cosecx})$

$= (tanx+cotx)(\frac{cosecx+secx}{secx.cosecx})$

$= (\frac{tanx}{secx.cosecx}+\frac{cotx}{secx.cosecx})(secx+cosecx)$

$= (sin^2x+cos^2x)(secx + cosec x)$

$= secx + cosec x$

= R.H.S.

Hence, proved.

Answer 2

Answer:

RHS

Step-by-step explanation:

L.H.S. = sinx (1+tanx) +cosx (1+cotx)sinx(1+tanx)+cosx(1+cotx)

= sinx(1+\frac{sin x}{cos x}) + cosx (1 +\frac{cosx}{sin x} )=sinx(1+

cosx

sinx

)+cosx(1+

sinx

cosx

)

= sin x(\frac{cos x+ sin x}{cos x} ) + cosx(\frac{sinx+cosx}{sinx} )=sinx(

cosx

cosx+sinx

)+cosx(

sinx

sinx+cosx

)

= tanx(cosx+sinx)+cotx(sinx+cosx)=tanx(cosx+sinx)+cotx(sinx+cosx)

= (tanx+cotx)(sinx+cosx)=(tanx+cotx)(sinx+cosx)

=(tanx+cotx)(\frac{1}{secx} + \frac{1}{cosecx})=(tanx+cotx)(

secx

1

+

cosecx

1

)

= (tanx+cotx)(\frac{cosecx+secx}{secx.cosecx})=(tanx+cotx)(

secx.cosecx

cosecx+secx

)

= (\frac{tanx}{secx.cosecx}+\frac{cotx}{secx.cosecx})(secx+cosecx)=(

secx.cosecx

tanx

+

secx.cosecx

cotx

)(secx+cosecx)

= (sin^2x+cos^2x)(secx + cosec x)=(sin

2

x+cos

2

x)(secx+cosecx)

= secx + cosec x=secx+cosecx

= R.H.S.

Hence, proved.