If the solution of the equation log cos x cot x+4 log sin x tan x=1, x ∈(0, (π/2)), is sin -1((α+√β/2)), where α, β are integers, then α+β is equal to :

Question

If the solution of the equation log cos x cot x+4 log sin x tan x=1, x ∈(0, (π/2)), is sin -1((α+√β/2)), where α, β are integers, then α+β is equal to : (A) 6 (B) 4 (C) 5 (D) 3

Tardigrade Admin · Accepted Answer

log cos x cot x+4 log sin x tan x=1 ⇒ ( ln cos x- ln sin x/ ln cos x)+4 ( ln sin x- ln cos x/ ln sin x)=1 ⇒( ln sin x)2-4( ln sin x)( ln cos x)+4( ln cos x)2=1 ⇒ ln sin x=2 ln cos x ⇒ sin 2 x+ sin x-1=0 ⇒ sin x=(-1+√5/2) ∴ α+β=4 Correct option (4)

Q. If the solution of the equation logcosx​cotx+4logsinx​tanx=1,x∈(0,2π​), is sin−1(2α+β​​), where α, β are integers, then α+β is equal to :

6

4

5

3

logcosx​cotx+4logsinx​tanx=1 ⇒lncosxlncosx−lnsinx​+4lnsinxlnsinx−lncosx​=1 ⇒(lnsinx)2−4(lnsinx)(lncosx)+4(lncosx)2=1 ⇒lnsinx=2lncosx ⇒sin2x+sinx−1=0⇒sinx=2−1+5​​ ∴α+β=4 Correct option (4)

Q. If the solution of the equation $lo g_{c o s x} cot x + 4 lo g_{s i n x} tan x = 1, x \in (0, \frac{π}{2})$ , is $sin^{- 1} (\frac{α + β}{2})$ , where $α$ , $β$ are integers, then $α + β$ is equal to :