Let's solve each trig identity step by step:
1. cosx + 1/sin^3x = cscx/(1 - cosx)
Starting with the left-hand side (LHS), you correctly converted the expression to a common denominator:
cosx + sin^3x/sin^3x
Now, we can combine the fractions:
(cosx * sin^3x + sin^3x) / sin^3x
Factoring out sin^3x from the numerator:
sin^3x (cosx + 1) / sin^3x
Canceling out sin^3x:
cosx + 1
Now, let's focus on the right-hand side (RHS):
cscx/(1 - cosx)
Using the reciprocal identity, cscx = 1/sinx:
(1/sinx) / (1 - cosx)
Multiplying the numerator and denominator by sinx:
1 / (sinx * (1 - cosx))
Now, let's simplify by multiplying sinx with 1 - cosx:
1 / (sinx - sinx * cosx)
Using the distributive property:
1 / (sinx - sinx * cosx)
Now, we can factor out sinx:
1 / (sinx * (1 - cosx))
Notice that the right-hand side (RHS) simplifies to the same expression obtained on the left-hand side (LHS):
cosx + 1
Hence, the trig identity is proven.
2. (sinx + cosx)(tanx + cotx) = secx + cscx
You correctly expanded the expression using the distributive property:
(sin^2x/cosx) + (sinxcosx/sinx) + (sinxcosx/cosx) + (cos^2x/sinx)
Now, let's simplify each term:
sin^2x/cosx simplifies to tanx using the definition of the tangent function.
sinxcosx/sinx simplifies to cosx using the canceling property of the sine function.
sinxcosx/cosx simplifies to sinx using the canceling property of the cosine function.
cos^2x/sinx simplifies to cotx using the definition of the cotangent function.
Therefore, the expression becomes:
tanx + cosx + sinx + cotx
Now, let's rewrite the right-hand side (RHS):
secx + cscx
Using the reciprocal identities, secx = 1/cosx and cscx = 1/sinx:
1/cosx + 1/sinx
Let's find a common denominator:
(sinx + cosx) / (sinx * cosx)
Therefore, the right-hand side (RHS) can be simplified to:
(sinx + cosx) / (sinx * cosx)
Notice that the LHS and RHS are equal:
tanx + cosx + sinx + cotx = (sinx + cosx) / (sinx * cosx)
Hence, the trig identity is proven.
Remember, when proving trig identities, it's important to simplify both sides of the equation using known trigonometric properties and identities until they match.