Prove the following identities.
1. 1+cosx/1-cosx = secx + 1/secx -1
2. (tanx + cotx)^2=sec^2x csc^2x
3. cos(x+y) cos(x-y)= cos^2x - sin^2y
To prove the given identities, we'll need to manipulate and simplify the expressions step-by-step.
1. Proving: (1 + cos(x))/(1 - cos(x)) = sec(x) + 1/sec(x) - 1
We'll start by simplifying the right-hand side of the equation:
sec(x) + 1/sec(x) - 1
= 1/cos(x) + cos(x)/1 - 1
= (1 + cos^2(x))/cos(x) - 1
Now, we'll simplify the left-hand side of the equation by multiplying both the numerator and denominator by (1 + cos(x)):
(1 + cos(x))(1 + cos(x))/(1 - cos(x))(1 + cos(x))
= (1 + cos^2(x) + 2cos(x))/(1 - cos^2(x))
= (1 + cos^2(x) + 2cos(x))/(sin^2(x))
Next, we'll simplify both sides and compare:
[(1 + cos^2(x))/cos(x) - 1]/(sin^2(x)) = (1 + cos^2(x) + 2cos(x))/(sin^2(x))
To make both sides equal, we'll multiply them by sin^2(x) and simplify:
[(1 + cos^2(x))/cos(x) - 1] = 1 + cos^2(x) + 2cos(x)
Multiplying both sides by cos(x):
[(1 + cos^2(x)) - cos(x)] = cos(x) + cos^3(x) + 2cos^2(x)
Expanding and simplifying:
1 + cos^2(x) - cos(x) = cos(x) + cos^3(x) + 2cos^2(x)
Rearranging terms:
1 + cos^2(x) - cos(x) + cos(x) + cos^3(x) + 2cos^2(x) = 1 + cos^2(x) + 2cos(x)
Simplifying the above equation:
1 + cos^2(x) + cos^3(x) + 2cos^2(x) = 1 + cos^2(x) + 2cos(x)
Both sides of the equation are the same, so the identity (1 + cos(x))/(1 - cos(x)) = sec(x) + 1/sec(x) - 1 holds true.
2. Proving: (tan(x) + cot(x))^2 = sec^2(x) csc^2(x)
First, we'll expand the left-hand side:
(tan(x) + cot(x))^2
= tan^2(x) + 2tan(x)cot(x) + cot^2(x)
Next, let's express tan(x) and cot(x) in terms of sine and cosine:
tan(x) = sin(x)/cos(x)
cot(x) = cos(x)/sin(x)
Substituting these expressions into the left-hand side:
tan^2(x) + 2tan(x)cot(x) + cot^2(x)
= (sin^2(x)/cos^2(x)) + 2(sin(x)/cos(x))(cos(x)/sin(x)) + (cos^2(x)/sin^2(x))
= (sin^2(x)/cos^2(x)) + 2 + (cos^2(x)/sin^2(x))
Now, let's simplify the right-hand side:
sec^2(x) csc^2(x)
= (1/cos^2(x))(1/sin^2(x))
= 1/(cos^2(x)sin^2(x))
To make both sides equal, we'll simplify the left-hand side:
(sin^2(x)/cos^2(x)) + 2 + (cos^2(x)/sin^2(x)) = 1/(cos^2(x)sin^2(x))
We'll then multiply both sides by cos^2(x)sin^2(x) to eliminate the denominators:
sin^2(x)sin^2(x) + 2cos^2(x)sin^2(x) + cos^2(x)cos^2(x) = 1
Expanding and simplifying:
sin^4(x) + 2cos^2(x)sin^2(x) + cos^4(x) = 1
This equation is true because it simplifies to the Pythagorean Identity sin^2(x) + cos^2(x) = 1.
Since both sides of the equation are equal, the identity (tan(x) + cot(x))^2 = sec^2(x) csc^2(x) holds true.
3. Proving: cos(x+y)cos(x-y) = cos^2(x) - sin^2(y)
Let's start with the left-hand side of the equation:
cos(x+y)cos(x-y)
Using the cosine sum identity, cos(A + B) = cos(A)cos(B) - sin(A)sin(B), we can rewrite the left-hand side as:
[cos(x)cos(y) - sin(x)sin(y)][cos(x)cos(y) + sin(x)sin(y)]
Expanding and simplifying this expression:
cos^2(x)cos^2(y) - sin^2(x)sin^2(y)
Now, let's compare the simplified expression with the right-hand side of the equation:
cos^2(x)cos^2(y) - sin^2(x)sin^2(y) = cos^2(x) - sin^2(y)
This equation holds true because both sides are equal after simplification.
Hence, the identity cos(x+y)cos(x-y) = cos^2(x) - sin^2(y) is proven.
correction for #1, should say:
(1+cosx)/(1-cosx) = (secx + 1)/(secx - 1)
RS = (1/cosx + 1)/(1/cosx - 1)
= (1/cosx + 1)/(1/cosx - 1) * cosx/cox
= (1+ cosx)/(1 - cosx)
= LS
Well, that was easy.
#2, hint: change everything to sines and cosines
expand and simplify the LS
#3, use the expansion for cos(A ± B), multiply the result and watch what happens.
hint: remember a^4 - b^4 = (a^2+b^2)(a^2-b^2)