Two shm are given by Y1 =40sinwt and Y2=10(sinwt+c.coswt). If their displacement are equal , then the value of c is

Y₂ = 10(sinwt + c.coswt)

 divide & multiply by (1+c²)^1/2

 Y₂ =10(1+c²)^1/2 [ sinwt(1/1+c²)^1/2 + (c²/1+c²)^1/2coswt ]

now put (1/1+c²)^1/2 = cos@ then

            (c²/1+c²)^1/2 = sin@

now Y₂ becomes

Y₂ = 10(1+c²)^1/2 [ sinwtcos@ + coswtsin@]

Y₂ = 10(1+c²)^1/2[sin(wt+@)]

Y₁ = 40sinwt

for same displacement , amplitude must be same so

 40 = 10(1+c²)^1/2

  c = 15^1/2

Approved