# If tan(cot θ) = cot(tan θ) then sinθ.cosθ = a.1/π b.π c.1/2π d.2/π π denotes pi.

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### 3 Answers

d. `2/pi`

`tan(cot(theta))=cot(tan(theta))`

`tan(cot(theta))=tan(pi/2-tan(theta))`

`cot(theta)=pi/2-tan(theta)`

`cot(theta)+tan(theta)=pi/2`

`cos(theta)/sin(theta)+sin(theta)/cos(theta)=pi/2`

`(cos^2(theta)+sin^2(theta))/(sin(theta)cos(theta))=pi/2`

`sin(theta)cos(theta)=2/pi`

Set x as angle

sen (ctg x)/ cos (ctg x) = cos(tg x)/ sen(tg x)\\ developing: sen (ctg x) sen(tg x) = cos(tg x) cos(ctg x) \\cos(tg x)cos(ctg x) - sen(tg x)sen(ctg x) = 0 \\This is the formula for the cosine addition of angles, where the angles are tg x and ctg x \\cos(tg x + cotg x)=0 \\ cos( sen x /cos x + cos x /sen x) = 0 \\ now: cosine is zero for x= pigrego/2 or x = 3/2 pigreco Therefore:\\ cos( 1 /senx cosx ) =0 \\1/senx cosx = pigreco /2 \\ 1/senx cosx = 3/2 pigreco\\ senx cosx = 2/pigreco\\ senx cosx = 2/3 pigreco\\ We can also write: sen 2x = 4/pigreco and sen 2x = 4/3 pigreco