Question

if sec theta -tan theta =x, show that sec theta + tan theta =1/x and hence find the value of cos theta and sin theta

Answer 1

The answer is given below :

Given that,

secθ - tanθ = x .....(i)

We know that,

sec²θ - tan²θ = 1

=> (secθ + tanθ)(secθ - tanθ) = 1

=> (secθ + tanθ) × x = 1

=> secθ + tanθ = 1/x

So, secθ + tanθ = 1/x [Proved]

secθ + tanθ = 1/x .....(ii)

Now, adding (i) and (ii), we get

2 secθ = x + 1/x

=> secθ = 1/2 (x² + 1)/x

=> cosθ = 2x/(x² + 1)

So, cosθ = 2x/(x² + 1)

Now,

sinθ = [√{(x² + 1)² - (2x)²}]/(x² + 1)

= (x² - 1)/(x² + 1)

Thank you for your question.

Answer 2

first rationalise sec theta - tan theta
then add equation 1 & 2 u will get the value of cos theta
subtract 1 & 2 for sin theta