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Question

If secθ=x+14x then prove that, secθ+tanθ=2x or 12x.


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Solution

Step 1: Find the possible values of tanθ using trigonometric identities

Given that: secθ=x+14x

1+tan2θ=sec2θtan2θ=sec2θ-1

On expanding, we get

tan2θ=x+14x2-1tan2θ=x2+116x2+12-1tan2θ=x2+116x2-12tan2θ=x-14x2tan2θ=±x-14x

Step 2: Prove the required result

When tanθ=x-14x we get,

secθ+tanθ=x+14x+x-14xsecθ+tanθ=2x

When tanθ=-x-14x we get,

secθ+tanθ=x+14x-x-14xsecθ+tanθ=12x

Hence proved that secθ+tanθ=12x and, secθ+tanθ=2x.


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