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Question

Assertion :nr=1(1+sec2rθ)=tan2nθcotθ Reason: cosθcos2θcos22θ....cos2n1θ=sin(2nθ)2nsinθ

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Assertion false but Reason is true
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Solution

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=12cos22r1θnr=1cos2rθ (1+cos2θ=2cos2θ)
=2n(nr=1cos2r1θ)2cos2θcos22θcos23θ....cos2nθ
Multiplying and dividing by cosθ
=cosθ[2n(cosθcos2θcos22θcos23θ....cos2n1θ)2cosθcos2θcos22θcos23θ....cos2n1θcos2nθ]
=cosθ[2n(cosθcos2θcos22θcos23θ....cos2n1θ)cos2nθ]
=cosθ[2n(sin(2nθ)2nsinθcos2nθ] (cosθcos2θcos22θ....cos2n1θ=sin2nθ2nsinθ)
=tan2nθcotθ
=R.H.S of Assertion.
Hence, assertion is true.
Now consider,cosθcos2θcos22θ....cos2n1θ
Multiplying and dividing by 2sinθ
=12sinθ(2sinθcosθcos2θcos22θ....cos2n1θ)
=12sinθ(sin2θcos2θcos22θ....cos2n1θ)
Now, multiply and divide by 2,
=122sinθ(sin22θcos22θ....cos2n1θ)
Again multiplying and dividing by 2,
=123sinθ(sin23θcos23θ....cos2n1θ)
Continuing this process, we get
=12nsinθ(sin2n1θcos2n1θ)
=sin(2nθ)2nsinθ
Hence, reason is true and is the correct explanation for assertion.

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