Question

if sec theta equals to 5 by 4 find the value of tan theta by 1 + tan theta multiplied by 1+ cot theta by cot theta​

Answer 1

QUESTION:

If secθ = 5/4 . Find the value of

${ \rm{ \frac{tan \theta}{ 1 + tan \theta} \times \frac{1 + cot \theta}{cot \theta} }}$

ANSWER :

$\setlength{\unitlength}{1mm}\begin{picture}(5,5)\put(0,0){\line(1,0){30}}\put(30,0){\line(0,1){40}}\put(0,0){\line(3,4){30}}\put(13,-4){3\ cm}\put(32,17){4\ cm}\put(8,23){5 cm}\end{picture}$

Given that

secθ = 5/4

In the above triangle

Hypotenuse = 5

Base = 4

Height = ?

By using Pythagoras theorem

(Hypotenuse)² = (Base)² + (Height)²

5² = 4² + H²

25 - 16 = H²

9 = H²

H = √9 = 3

Here,

tanθ = opposite/adjacent = 3/4

cotθ = adjacent/opposite = 4/3

Now,

${ \rm{ = \dfrac{ \frac{3}{4} }{1 + \frac{3}{4} } + \dfrac{ \frac{4}{3} }{1 + \frac{4}{3} } }} \\ \\ \dfrac{ \frac{3}{4} }{ \frac{7}{4} } + \frac{ \frac{4}{3} }{ \frac{7}{3} } = \frac{3}{7} + \frac{4}{7} = \frac{7}{7} = 1$