2a Prove cotx1 cotx1tanx 1tanx 2bProve 1 1sinx 1 1sinx2

2a.) Prove: cot(x)+1 / cot(x)=1+tan(x) / 1-tan(x)

2b.)Prove: 1 / 1-sin(x) - 1 / 1+sin(x)=2sec(x)tan(x)

Please show all your work.

Solution

2a) this question should be:

cot(x) + 1 / ( cot x - 1) = 1+tan(x) / 1-tan(x)

LHS:

cot(x) + 1 / ( cot x - 1)

=[ (1/tan x) + 1 ] / [ ( 1/tanx) - 1]

= 1+tanx / 1- tan x

= RHS

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

LHS:

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

= 1+sin x - ( 1- sin x) / [ ( 1- sin x)* ( 1+ sin x)]

= 2*sin x / 1- sin² x

= 2*sin x / cos² x [ from sin²x + cos²x = 1]

= 2*tan x* sec x [ sin x/ cos x = tan x, 1/ cos x = sec x)

= RHS