Find the equations of all the tangents to the curve y = cos(x + y), -2π ≤ x ≤ 2π, that are parallel to the line x + 2y = 0. (CBSE 2016,2017)

y = cos(x + y)

we have to find equation of tangent which is parallel to x + 2y = 0

Slopes of parallel lines are equal hence slope of line x + 2y = 0 will be the slope of tangent

Let us find slope of line by writing it in form y = mx + c

⇒ x + 2y = 0

⇒ 2y = -x

⇒ y = -(1/2)x

Hence slope of line is -(1/2) and hence slope of tangent is -(1/2)

Slope of tangent is given by hence

Now to find equation of tangent we need point on curve where the slope is -(1/2)

Differentiate given y with respect to x

Put slope

⇒ 1 + sin(x + y) = 2sin(x + y)

⇒ sin(x + y) = 1

Hence x + y = 90°

Put this value of (x + y) in curve equation y = cos(x + y)

⇒ y = cos90°

⇒ y = 0

Put y = 0 in sin(x + y) = 1

⇒ sinx = 1

Now given that x is from -2π to 2π hence we get sinx = 1 for two values of x

One is and other is

Hence we have two points on curve where slope of tangent is -(1/2)

Those points are and and hence we have 2 tangents

Writing equation of those tangents using slope point form

Tangent passing through and having slope -1/2

⇒ 4y = -2x + π

⇒ 2x + 4y = π

Tangent passing through and having slope -1/2

⇒ 4y = -2x – 3π

⇒ 2x + 4y = -3π

Hence equations of tangents to curve y = cos(x + y) that are parallel to the line x + 2y = 0 are 2x + 4y = π and 2x + 4y = -3π