Solveeit Logo
Login

Question

Question: If \(\sec\theta = 1\frac{1}{4}\), then \(\tan\frac{\theta}{2} =\)...

If sec⁡θ=114\sec\theta = 1\frac{1}{4}, then tan⁡θ2=\tan\frac{\theta}{2} =

A

13\frac{1}{3}

B

34\frac{3}{4}

C

14\frac{1}{4}

D

54\frac{5}{4}

Answer

13\frac{1}{3}

Explanation

Solution

Given that sec⁡θ=54\sec\theta = \frac{5}{4}

sec⁡θ=1+tan⁡2(θ/2)1−tan⁡2(θ/2)⇒54=1+tan⁡2(θ/2)1−tan⁡2(θ/2)\sec\theta = \frac{1 + \tan^{2}(\theta/2)}{1 - \tan^{2}(\theta/2)} \Rightarrow \frac{5}{4} = \frac{1 + \tan^{2}(\theta/2)}{1 - \tan^{2}(\theta/2)}

⇒ 5−5tan⁡2(θ/2)=4+4tan⁡2(θ/2)5 - 5\tan^{2}(\theta/2) = 4 + 4\tan^{2}(\theta/2)

⇒ 9tan⁡2(θ/2)=1⇒tan⁡(θ/2)=139\tan^{2}(\theta/2) = 1 \Rightarrow \tan(\theta/2) = \frac{1}{3}.