Prove that : sin x ( 1+ tan x) + cos x (1+ cot x) = sec x + cosec x
Answers
Step-by-step explanation:
I hope this will help you
have to prove that, sinx(1+tanx) +cosx(1+cotx) =secx+cosecx
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.