To find: Differentiation of (tan x + sec x) (cot x + cosec x)

Formula used: (i) (uv)′ = u′v + uv′ (Leibnitz or product rule)

(ii)

(iii)

(iv)

(v)

Let us take u = (tan x + sec x) and v = (cot x + cosec x)

Putting the above obtained values in the formula:-

(uv)′ = u′v + uv′

[(tan x + sec x) (cot x + cosec x)]’

= [secx (secx + tanx)] × [(cot x + cosec x)] + [(tan x + sec x)] × [-cosecx (cosecx + cotx)]

= (secx +tanx) [secx(cotx + cosecx) - cosecx(cosecx + cotx)]

= (secx + tanx) (secx – cosecx) (cotx + cosecx)

Ans) (secx + tanx) (secx – cosecx) (cotx + cosecx)