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Question

If (1-tan2θ)sec2θ=12, then the general value of θ is


A

±π6

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B

nπ+π6

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C

2±π6

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D

None of these

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Solution

The correct option is A

±π6


Explanation of the correct option:

Step-1. Calculate the value of tanθ:

Given,(1-tan2θ)sec2θ=12,

(1-tan2θ)(1+tan2θ)=12[sec2(θ)-tan2(θ)=1sec2(θ)=1+tan2(θ)]2(1-tan2θ)=1(1+tan2θ)2-2tan2θ=1+tan2θ2-1=2tan2θ+tan2θ1=3tan2θtan2θ=13tanθ=±13

Step2. Find the value of θ:

We know that

if tan(θ)=tan(α) then the general solution of θ=+α,nZ.

We have tanθ=±13

tanθ=±tanπ6tan(π6)=13θ=nπ±π6

Hence, Option(A) is the correct answer.


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