Verify this Trig Identiity

Question

Problem: (1/sec(x)+1) - (1/sec(x)-1) = ( -2cot^2 (x) )

Answer 1

prove (1/[sec+1])-(1/[sec-1])=-2cot^2

(1/[sec+1])-(1/[sec-1])=

(sec-1-sec-1)/(sec^2-1)=

-2/[sec^2-1]=

1+tan^2=sec^2 is a trig identity

so, tan^2=sec^2-1

substitute

-2/tan^2=

cot=1/tan is a trig identity

so, cot^2=1/tan^2

now -2tan^2=-2(1/tan^2)

substitute again

-2cot^2=-2cot^2

Answer 2

(1 / sec x + 1) - (1 / sec x - 1)

[(sec x - 1) - (sec x + 1)] / [(secx + 1)(secx - 1)]

-2 / sec² - 1

-2 / tan² x

-2 cot² x

QED

Answer 3

1/(sec x + 1) - 1/(sec x - 1) --> in order to subtract, there must be a common denominator, so we must multiply the numerator and denominator of 1/(sec x + 1) by (sec x - 1) and we must multiply the numerator and denominator of 1/(sec x - 1) by (sec x + 1). Once the denominators are the same, the resulting subtraction will look like...

[(sec x - 1) - (sec x + 1)] / (sec² x - 1) --> the sec x and -sec x in the numerator cancel each other out so...

= -2/(sec² x - 1) --> given the pythagorean identity 1 + tan² x = sec² x, we can rearrange this so that sec² x - 1 = tan² x, so we get...

= -2/tan² x --> 1/tanx is the same as cot x, therefore...

= -2cot² x

Answer 4

1/(sec+1) - 1/(sec-1)

substitute 1/cos for sec

1/(1/cos + 1) - 1/(1/cos - 1)

will lead to

cos/(1 + cos) - cos/(1 - cos)

which leads to

cos * (1-cos) - cos * (1+ cos) / (1-cos^2)

combining to a single fraction you get

(cos - cos^2 - cos - cos^2 - cos) / (1 - cos^2)

simplifying you get (and since sin^2 + cos^2 = 1)

-2 * cos ^ 2 / sin ^2

which equals -2 * cot^2