Question

If sin theta cos theta and tan theta are in gp then the value of cot^6theta

Answer 1

sinθ , cosθ , tanθ are in GP
We know, in Geometric progression , common ratio is always constant .
Means ratio of two consecutive terms is always same. Like a₁ , a₂, a₃ are in GP then, a₂/a₁ = a₃/a₂ = constant

So, cosθ /sinθ= tanθ/cosθ
cos²θ = tanθ.sinθ = (sinθ/cosθ) sinθ
cos²θ = sin²θ/cosθ
cos³θ = sin²θ

Now, cot⁶θ = cos⁶θ/sin⁶θ = cos⁶θ/(sin²θ)³ = cos⁶θ/(cos³θ)³
= (cos³θ)²/(cos³θ)³ = 1/cos³θ or, 1/sin²θ
Hence , answer is cosec²θ or sec³θ

Answer 2

Answer:

sinA, cosA, tanA arein GP

so, cos^2A=sinA×tanAcos2A

=sin2AcosAcos2Asin2A

=1cosAcot2A=secA

Now squaring both side we get, cot4A=sec2Acot4A

=1+tan2Acot4A

=1+1cot2Acot6A

=cot2A+1

So cot6− cot2A=1