Question

Example 5. If y =
COS X + sinx
TT
prove that
dy
dx
sec2
+
COS X - sin x
4​

Answer 1

Answer:

dy

=sin(x+y)+cos(x+y)

Put u=x+y⇒y=u−x⇒

dx

dy

=

dx

du

−1

∴

dx

du

−1=sinu+cosu⇒

dx

du

=(1+cosu)+sinu

⇒

dx

du

=2cos

2

2

u

+2sin

2

u

cos

2

u

⇒

2

1

sec

2

2

u

dx

du

=1+tan

2

u

⇒

2

1

⎝

⎛

tan

2

u

+1

sec

2

2

u

⎠

⎞

du=dx

Integrate on both sides

⇒log(1+tan

2

u

)=x+c

⇒log(1+tan

2

x+y

)=x+c