trigonometry identity question?

Question

I am stuck at this question, can someone please explain how to find this?

1 - sin x
---------- = (sec x - tan x) ²
1 + sin x

Answer 1

Working on the left side....multiply numerator and denominator
by 1 - sin x

This gives (1 - sin x)^2/ (1 - sin x + sin x - sin^2 x)

Simplifying to (1 - sin x)^2 / cos^2 x

The right side goes to (1/cos x -sin x/cosx)^2

Simplifying to {(1 -sin x /cos x)^2

Expanding to (1 - sin x)^2/ cos^2 x

Left side = right side

Answer 2

(1 - sin x)/(1 + sin x)

Multiply numerator and denominator by (1 - sin x) and get:
(1 - sin x)^2/(1 - sin^2 x) = (1 - sin x)^2/cos^2 x
= [(1 - sin x)/cos x]^2 = (sec x - tan x)^2

Answer 3

(1 - sin(x))/(1 + sin(x)) = (sec(x) - tan(x))^2

(sec(x) - tan(x))^2 = ((1/cos(x)) - (sin(x)/cos(x)))^2 = ((1 - sin(x))/cos(x))^2

(1 - sin(x))/(1 + sin(x)) = ((1 - sin(x))/cos(x))^2

cross multiply

(1 - sin(x))cos(x)^2 = (1 + sin(x))(1 - sin(x))^2

(1 - sin(x))(1 - sin(x)^2) = (1 - sin(x))^2 * (1 - sin(x))

divide both sides by (1 - sin(x))^2

1 - sin(x) = 1 - sin(x)

Don't feel bad, sometimes it takes me a while to figure out how to go about doing this until it works out right.

Answer 4

Multiply both the numerator and denominator by (1-sinx.)Note that
(1+sinx)(1-sinx)=(1-sin^2x) {This is like (a+b)(a-b)=a^2-b^2}
(1-sin^2x)=cos^2x {as cos^2x+sin^2x=1}
So the Left Hand side becomes (1-sinx)^2/cos^2x
=(1/cosx-sinx/cosx)^2
=(secx-tanx)^2 {1/cosx=secx, sinx/cosx=tanx}
Hence the result

Answer 5

1 - sin x
---------- = (sec x - tan x) ²
1 + sin x

take the left side of the eqn;
(1-sinx)(1-sinx)/(1+sinx)(1-sinx)

=(1-sinx)^2/1-(sinx)^2;

----1-(sinx)^2 is cos(x)^2---

=1-2sinx+(sinx^2)/(cosx)^2

=(1/(cosx)^2)-(2sinx/(cosx)^2)+((sinx^2)/(cosx)^2)

=(secx)^2-2secx.tanx+(tanx)^2

=(sec x - tan x)^2=Right hand side!!!

Answer 6

1 - sin x
---------- = (sec x - tan x) ²
1 + sin x

LHS =

1 - sin x
----------- =
1 + sin x

1 - sin x ... 1 - sin x
----------- X ------------ =
1 + sin x ... 1 - sin x

1 - 2 sinx + sin²x
---------------------- =
1 - sin²x

1 - 2 sinx + sin²x
---------------------- =
cos²x

1 ..... ........... 1 ....... sinx ... sin²x
------- – 2 X ------- X ------ + ------- =
cos²x ........ cosx ... cosx .. cos²x

sec²x - 2secxtanx + tan²x =

(secx - tan x)²

= RHS .... QED