If sec θ = 5 4 , then tan θ 2 = Options: A . &nbsp 1 3 B . &nbsp 3 4 C . &nbsp 1 4 D . &nbsp 1 9

Answer: Option A : A Given that sec θ = 5 4 sec θ = 1 + t a n 2 ( θ 2 ) 1 − t a n 2 ( θ 2 ) ⇒ 5 4 = 1 + t a n 2 ( θ 2 ) 1 − t a n 2 ( θ 2 ) ⇒ 5 − 5 t a n 2 ( θ 2 ) = 4 + 4 t a n 2 ( θ 2 ) ⇒ 9 t a n 2 ( θ 2 ) = 1 ⇒ t a n ( θ 2 ) = 1 3 .

If secθ = 54, then tan θ2 =

Discussion Forum : Trigonometric Functions

Question -

If sec $θ$ = $\frac{5}{4}$ , then tan $\frac{θ}{2}$ =

Options:

A .

\frac{1}{3}

B .

\frac{3}{4}

C .

\frac{1}{4}

D .

\frac{1}{9}

Answer: Option A
:
A

Given that sec $θ$ = $\frac{5}{4}$

sec $θ$ = $\frac{1 + t a n^{2} (\frac{θ}{2})}{1 - t a n^{2} (\frac{θ}{2})}$ $\Rightarrow$ $\frac{5}{4}$ = $\frac{1 + t a n^{2} (\frac{θ}{2})}{1 - t a n^{2} (\frac{θ}{2})}$

$\Rightarrow$ $5 - 5 t a n^{2} (\frac{θ}{2}) = 4 + 4 t a n^{2} (\frac{θ}{2})$

$\Rightarrow$ $9 t a n^{2} (\frac{θ}{2})$ = 1 $\Rightarrow$ $t a n (\frac{θ}{2})$ = $\frac{1}{3}$ .

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