The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
∏nr=1(1+sec2rθ)
=∏nr=1(1+cos2rθcos2rθ)
=∏nr=12cos22r−1θ∏nr=1cos2rθ (∵1+cos2θ=2cos2θ)
=2n(∏nr=1cos2r−1θ)2cos2θcos22θcos23θ....cos2nθ
Multiplying and dividing by cosθ
=cosθ[2n(cosθcos2θcos22θcos23θ....cos2n−1θ)2cosθcos2θcos22θcos23θ....cos2n−1θcos2nθ]
=cosθ[2n(cosθcos2θcos22θcos23θ....cos2n−1θ)cos2nθ]
=cosθ[2n(sin(2nθ)2nsinθcos2nθ] (∵cosθcos2θcos22θ....cos2n−1θ=sin2nθ2nsinθ)
=tan2nθcotθ
=R.H.S of Assertion.
Hence, assertion is true.
Now consider,cosθcos2θcos22θ....cos2n−1θ
Multiplying and dividing by 2sinθ
=12sinθ(2sinθcosθcos2θcos22θ....cos2n−1θ)
=12sinθ(sin2θcos2θcos22θ....cos2n−1θ)
Now, multiply and divide by 2,
=122sinθ(sin22θcos22θ....cos2n−1θ)
Again multiplying and dividing by 2,
=123sinθ(sin23θcos23θ....cos2n−1θ)
Continuing this process, we get
=12nsinθ(sin2n−1θcos2n−1θ)
=sin(2nθ)2nsinθ
Hence, reason is true and is the correct explanation for assertion.