Im really struggling with these proving identities problems can somebody please show me how to do these? I'm only aloud to manipulate one side of the equation and it has to equal the other side of the equation at the end
Problem 1. Sinx/(cotx+1) + cosx/(tanx+1) = 1/(sinx+cosx)
Problem 2. sinx + cosx + sinx + tanx + cosxcotx = secx + cscx
Problem 3. ((sinx + cosx)/(1 + tanx))^2 + ((sinx - cos^2x)/(1 - cotx))^2 = 1
Problem 4. ((1 + sinx)/cosx) + (cosx/(1 + sinx)) = 2secx
often it's easier to work with just sin and cos.
working just on the left side, we have
sin/(cot+1) + cos/(tan+1)
sin/(cos/sin+1) + cos/(sin/cos+1)
sin^2/(cos+sin) + cos^2(sin+cos)
(sin^2 + cos^1)/(sin+cos)
1/(sin+cos)
ta-daaaah
2. I think you have a typo , it should have been
sinx + cosx + sinxtanx + cosxcotx = secx + cscx
LS = sinx + cosx + sinx(sinx/cosx) + cosx(cosx/sinx
using a LCD of sinxcosx
= (sin^2x cosx + sinxcos^2x) + sin^3 x + cos^3 x)/(sinxcos)
= (cosx(sin^2 x + cos^2 x) + sinx(sin^2 x + cos^2 x) )/(sinxcosx)
= ( cosx (1) + sinx (1) )/(sinxcosx)
= cosx/(sinxcosx) + sinx/(sinxcosx)
= 1/sinx + 1/cosx
= cscx + secx
= RS
try the others, following Steve's suggestion of changing all into sines and cosines
thank you so much
Prove 1-(sinxtanx)/(1+secx)
To solve these proving identities problems, you'll need to use trigonometric identities and properties to manipulate the expressions on one side of the equation until it matches the other side. Here's how you can approach each of the problems:
Problem 1:
Start with the left side of the equation and manipulate it to get the right side:
1. Combine the fractions on the left side by finding a common denominator for cot(x) + 1 and tan(x) + 1.
Solution:
1. Multiply the first fraction (Sin(x) / (cot(x) + 1)) by [(tan(x) + 1) / (tan(x) + 1)] and the second fraction (cos(x) / (tan(x) + 1)) by [(cot(x) + 1) / (cot(x) + 1)] to find a common denominator.
2. Simplify the resulting expression and combine the fractions.
3. Manipulate and simplify the expression until it matches the right side of the equation (1 / (sin(x) + cos(x))).
Problem 2:
Start with the left side of the equation and manipulate it to get the right side:
1. Combine similar terms on the left side, such as sin(x) and sin(x), cos(x) and cos(x), and so on.
2. Use the trigonometric identities sec(x) = 1/cos(x) and csc(x) = 1/sin(x) to express sec(x) and csc(x) in terms of cos(x) and sin(x).
3. Manipulate and simplify the expression until it matches the right side of the equation (sec(x) + csc(x)).
Problem 3:
Start with the left side of the equation and manipulate it to get the right side:
1. Simplify the expressions within the parentheses on the left side by using the trigonometric identities tan(x) = sin(x)/cos(x) and cot(x) = cos(x)/sin(x).
2. Combine the two terms on the left side of the equation.
3. Manipulate and simplify the expression until it matches the right side of the equation (1).
Problem 4:
Start with the left side of the equation and manipulate it to get the right side:
1. Find a common denominator for the two fractions on the left side by multiplying the first fraction by (cos(x) + 1) / (cos(x) + 1) and the second fraction by (1 + sin(x)) / (1 + sin(x)).
2. Simplify and combine the fractions on the left side.
3. Use the trigonometric identity sec(x) = 1/cos(x) to manipulate and simplify the expression until it matches the right side of the equation (2sec(x)).
Remember to carefully apply trigonometric identities, simplify expressions, and perform algebraic manipulations to both sides of the equation until they are equal.