Question

sec theta - tan theta = x then the value of sin theta

Answer 1

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Answer 2

Hi...☺

Here is your answer...✌

GIVEN THAT,

secθ - tanθ = x -----(1)

We know that

sec²θ - tan²θ = 1

(secθ + tanθ) (secθ - tanθ) = 1

(secθ + tanθ) x = 1

secθ + tanθ = 1/x -----(2)

Adding eq(1) and (2)
We get,

$2 \sec \theta = x + \frac{1}{x} \\ \\ \sec \theta = \frac{ {x}^{2} + 1 }{2x}$

Subtracting eq(1) from (2)
We get,

$2 \tan \theta = \frac{1}{x} - x \\ \\ \tan \theta = \frac{1 - {x}^{2} }{2x}$

Now,

$\sin \theta = \frac{ \tan \theta }{ \sec \theta } \\ \\ \sin \theta = \frac{ \frac{1 - {x}^{2} }{2x} }{ \frac{1 + {x}^{2} }{2x} } \\ \\ \sin \theta = \frac{1 - {x}^{2} }{1 + {x}^{2} }$