1) 1+cos(3t)/ sin(3t) + sin(3t)/( 1+ cos(3t))= 2csc(3t)
2) sec^2 2u-1/ sec^2 2u= sin^2 2u
3) cosB/1- sinB= secB+ tanB
Please type your equations using brackets to show the proper order of operation.
The way you typed them, none are identities.
e.g.
for the last one you probably meant:
cosB/(1-sinB) = secB + tanB
then:
RS = 1/cosB + sinB/cosB
= = (1-sinB)/cosB * (1+sinB)/(1+sinB)
= (1 - sin^2 B)/(cosB(1+sinB)
= cos^2B/(cosB(1+sinB))
= cosB/(1+sinB)
= LS
1)1+cos(3t)/ sin(3t) + sin(3t)/( 1+ cos(3t))= 2csc(3t)
2)sec^2 (2u-1)/ sec^2 (2u)= sin^2 (2u)
in 1) since all angles are 3t, I will let x = 3t for easier typing
Again, the way you typed it, it does not work.
You must have meant
(1+cosx)/sinx + sinx/(1+cosx) = 2cscx
LS = ((1+cosx)^2 + sin^2x)/(sinx(1+cosx))
= (1 + 2cosx + cos^2x + sin^2x)/(sinx(1+cosx))
= (2+2cosx)/(sinx(1+cosx))
= 2(1+cosx)/(sinx(1+cosx))
= 2/sinx
= 2cscx, but x = 3t
= 2csc(3t)
= RS
2) does not work the way you typed it.
(sin^2 è+ cos^2 è)^3 =1
(sin^2 è+ cos^2 è)^3 =1
That is much too easy!!!!
what is the value of sin^2 è+ cos^2 è ?
To solve these equations, we will manipulate both sides of the equations individually to simplify them and make them equal to each other.
1) 1+cos(3t)/sin(3t) + sin(3t)/(1+cos(3t)) = 2csc(3t)
Step 1: Find common denominators for the fractions.
Multiply the first fraction by (1+cos(3t))/(1+cos(3t)).
Multiply the second fraction by sin(3t)/sin(3t).
(1+cos(3t))(1+cos(3t))/(sin(3t)(1+cos(3t))) + (sin(3t))(sin(3t))/(sin(3t)(1+cos(3t))) = 2csc(3t)
Step 2: Combine the fractions over the common denominator.
(1+cos(3t))(1+cos(3t)) + (sin(3t))(sin(3t)) = 2csc(3t)
Step 3: Expand and simplify.
1 + 2cos(3t) + cos^2(3t) + sin^2(3t) = 2csc(3t)
Step 4: Apply the Pythagorean identity sin^2(θ) + cos^2(θ) = 1.
1 + 2cos(3t) + 1 = 2csc(3t)
2 + 2cos(3t) = 2csc(3t)
Step 5: Divide both sides by 2.
1 + cos(3t) = csc(3t)
Step 6: Apply the reciprocal identity csc(θ) = 1/sin(θ).
1 + cos(3t) = 1/sin(3t)
Step 7: Apply the Pythagorean identity sin^2(θ) + cos^2(θ) = 1.
sin^2(3t) + cos^2(3t) + cos(3t) = 1
Step 8: Rearrange the terms.
cos^2(3t) + cos(3t) + sin^2(3t) = 1
Step 9: Simplify.
cos(3t) + 1 = 1
Step 10: Subtract 1 from both sides.
cos(3t) = 0
So the solution to this equation is cos(3t) = 0.
2) sec^2(2u) - 1/sec^2(2u) = sin^2(2u)
Step 1: Rewrite the secant squared terms as 1/cosine squared.
(1/cos^2(2u)) - (1/(1/cos^2(2u))) = sin^2(2u)
Step 2: Simplify the double reciprocal in the second term.
(1/cos^2(2u)) - cos^2(2u) = sin^2(2u)
Step 3: Find a common denominator for the fractions.
Multiply the first fraction by cos^2(2u)/cos^2(2u).
(1/cos^2(2u))(cos^2(2u)/cos^2(2u)) - cos^2(2u) = sin^2(2u)
Step 4: Combine the fractions over the common denominator.
cos^2(2u)/cos^2(2u) - cos^2(2u) = sin^2(2u)
Step 5: Simplify.
1 - cos^2(2u) = sin^2(2u)
Step 6: Apply the Pythagorean identity sin^2(θ) + cos^2(θ) = 1.
1 - cos^2(2u) = 1 - cos^2(2u)
This equation is an identity, which means that it holds true for all values of u.
3) cos(B)/(1 - sin(B)) = sec(B) + tan(B)
Step 1: Multiply both sides of the equation by (1 - sin(B)).
[cos(B)/(1 - sin(B))](1 - sin(B)) = (sec(B) + tan(B))(1 - sin(B))
Step 2: Simplify the left side of the equation.
cos(B) = (sec(B) + tan(B))(1 - sin(B))
Step 3: Expand the right side of the equation.
cos(B) = sec(B) + tan(B) - sec(B)sin(B) - tan(B)sin(B)
Step 4: Rearrange the terms.
cos(B) = sec(B) - sec(B)sin(B) + tan(B) - tan(B)sin(B)
Step 5: Apply the identity sec(B) - sec(B)sin(B) = 1.
cos(B) = 1 + tan(B) - tan(B)sin(B)
Step 6: Factor out tan(B) from the last two terms.
cos(B) = 1 + tan(B)(1 - sin(B))
Step 7: Apply the Pythagorean identity 1 - sin^2(B) = cos^2(B).
cos(B) = 1 + tan(B)cos^2(B)
Step 8: Multiply both sides by cos(B).
cos^2(B) = cos(B) + tan(B)cos^2(B)
Step 9: Subtract cos(B) from both sides.
cos^2(B) - cos(B) = tan(B)cos^2(B)
Step 10: Divide both sides by cos^2(B).
1 - cos(B) = tan(B)
This equation is an identity that holds true for all values of B except for B = kπ + π/2, where k is an integer.