Question

find the value of theta .. if cos theta /1-sin theta + cos theta / 1+ sin theta =4?

Answer 1

cos theta/1-sin theta+cos theta/1+sin theta

cos theta (1+sin theta)+cos theta(1-sin theta)/1¹-sin²theta

cos theta+cos theta sin theta +cos theta-sin theta cos theta/1-sin² theta

cos theta+cos theta/cos² theta

2cos theta/cos²theta

2/cos theta=4

cos theta=1/2

theta=60° _____________(as cos 60°=1/2)

Answer 2

θ = 60°  or 2nπ ± 60° if

$\dfrac{\cos \theta}{1- \sin \theta} +\dfrac{\cos \theta}{1+ \sin \theta}=4$

Given:

$\dfrac{\cos \theta}{1- \sin \theta} +\dfrac{\cos \theta}{1+ \sin \theta}=4$

To Find :

Value of θ

Solution:

Step 1:

Take LCM in denominator and simplify

$\dfrac{\cos \theta (1+ \sin \theta) + \cos \theta (1- \sin \theta)}{(1- \sin \theta)(1+ \sin \theta)}=4$

Step 3:

Take cosθ  common in numerator and simplify

$\dfrac{\cos \theta (1+ \sin \theta+1-\sin \theta)}{(1- \sin \theta)(1+ \sin \theta)}=4$

$\dfrac{\cos \theta (2)}{(1- \sin \theta)(1+ \sin \theta)}=4$

Step 4:

Use identity (a + b)(a - b) = a² - b²  where a = 1  b = sinθ

$\dfrac{2\cos \theta }{1- \sin^2 \theta}=4$

Step 5:

Use identity  $1- \sin^2 \theta = \cos^2 \theta$ and simplify

$\dfrac{2\cos \theta }{ \cos^2 \theta}=4$

$\dfrac{2 }{ \cos \theta}=4$

Step 6:

Solve for cosθ

cosθ = 2/4

cosθ = 1/2

Step 7:

Solve for  θ   using cos60° = 1/2  and general solution for  cosx = cosα is

x = 2nπ ± α

θ = 60°   or 2nπ ± 60°