If y = 2cos 2θ + cos θ / cos 3θ + cos^2 θ + cos θ Then value of y'' + y' + y is

Question

If \(y = \frac {2cos\,2\theta + cos \theta}{cos \,3\theta +cos ^2 \theta + cos \,\theta}\) Then value of y'' + y' + y is

(1) secθ(1 – tan³θ) 

(2) tanθ(sec³θ + 2tan²θ) 

(3) secθ(2sec²θ + tanθ) 

(4) cotθ(sec³θ + 2tanθ)

Answer 1

Correct option is (3) secθ(2sec²θ + tanθ) 

y = secθ 

y′ = secθ tanθ 

y′′ = sec³θ + tanθ⋅(secθ tanθ) 

= sec³θ + sec tan²θ

y′′ + y′ + y = sec³θ + secθtan²θ + secθ tanθ + secθ 

= secθ (sec²θ + 1) + secθ tanθ(tanθ + 1) 

= secθ (sec²θ + 1 + tan2θ + tanθ) 

= secθ (2sec²θ + tanθ)