Proving some identities!?

Question

For now * = theta
A. tan * + cot * = sec*csc*
B. 1-2sin^2* = 2cos*-1
C. (cos * - sin*)^2 + 2sin*cos* = 1
D. cos^4* - sin^4* = cos^2* - sin^2*
E. cos^ 2* - sin^2* = 1-tan^2* / 1+tan^2*
F. (sec* - tan *)^2 = 1- sin* / 1+sin*
G.(tan * / 1- cot*) + (cot* / 1- tan*) = 1+tan* + cot*
H. (1+tan* / 1-tan*) = (cos* + sin* / cos* - sin*)
I. (sin* / csc* - 1) - (sin* / cot^2) = tan^2*

Try and answer...thnx!

Answer 1

For clarity allow my use of x instead of theta.
A. tan x + cot x = (sin x/cos x) + (cos x/sin x)
= (sin^2 x+cos^2 x) / [(sin x).(cos x)]
= 1/[(sin x).(cos x)] = cosec x . sec x

B. I think you've made a mistake here, it should be
1- 2sin^2 x = 2cos^2 x -1
then,
1 - 2sin^2 x = 1 - 2(1-cos ^2 x) = 2cos^2 x -1

C. (cos x - sin x)^2 + 2sin x cos x
= cos^2 x -2cos x sin x + sin^2 x + 2sin x cos x
= cos^2 x + sin^2 x =1
D. cos^4 x - sin^4 x = (cos^2 x - sin^2 x).(cos^2 x + sin^2 x)
= (1) (cos^2 x - sin^2 x)
E. cos^2 x -sin^2 x = (1 - tan^2 x) .cos^2 x
by dividing by cos^2 x and then multiplying so that the
expression is unchanged on the LHS.
= (1 - tan^2 x) / sec^2 x = (1-tan^2 x) / (1+tan^2 x)
F. (sec x - tan x)^2 = sec^2 x - 2sec x.tan x + tan^2 x
= (1/cos^2 x) - (2sin x / cos^2 x) + (sin^2 x/cos^2 x)
= [1-2sin x+sin^2 x]/(1-sin^2 x)
= (1-sin x)^2 / (1-sin x)(1+sin x) = (1-sin x)/(1+sin x)
G. Henceforth I shall give instructions,
make a common denominator and simplify noting that
cot x.tan x = 1
H. Note that tan x = sin x / cos x and making cos x the common
denominator cancel it out in the top and bottom.
I. Note that csc x = 1/sin x and that cot x = cos x / sin x = 1/tan x

These should allow you to prove the rest. Hope this helps!

Answer 2

A)
tan t + cot t = sin t/cos t+ cost/sin t =
= (sin^2t+cos^2t)/cost sin t = 1/costsint = 1/cost . 1/sint = sec t cosec t
B)
1 - 2 sin^2 t = 1-2(1- cos ^2 t) = 2 cos^2 t - 1
C)
(cos t - sin t)^2 = cos^2 t + sin^ 2 t - 2 sin t cos t = 1- 2 sin t cos t
so (cos t - sin t)^2 + 2 sin t cos t =1
D)
Cos ^ 4 t - sin ^ 4 t = (cos ^2 t)^2 - (sin ^2 t)^2
= (cos^2 t+ sin^2 t)(cos ^2 t- sin^2 t) = (cos^2 t - sin ^2 t)

You should be able to work out the rest

Answer 3

When people want to use certain symbols in Yahoo! Answers like theta, why don't you use the actual symbols:

Theta - θ
Pi - π

Answer 4

LHS = a million/(cosec x + cot x) + a million/(cosec x - cot x) = (cosec x - cot x) + (cosec x - cot x) / (cosec x - cot x)(cosec x + cot x) = 2cosec x / (cosec^2 x - cot^2 x) = 2cosec x/ a million (utilizing reality a million + cot^2 x = cosec^2 x) = 2cosec x

Answer 5

do you really expect us to do your hw?
everything???
you can answer one two questions... understand it and then to do the rest be your self!!
this is just rude

Answer 6

tis your homework? solve on ur own!sorry