1) What is equivalent to [tex]\frac{cos(x)}{1 + sin(x)}[/tex]?
Multiply the numerator and denominator by 1 - sin(x):
[tex]\frac{cos(x)}{1 + sin(x)}[/tex] * [tex]\frac{1 - sin(x)}{1 - sin(x)}[/tex]
[tex]\frac{cos(x)(1 - sin(x))}{1 - sin^{2}(x)}[/tex]
Use this trigonometric identity
cos²(x) + sin²(x) = 1
1 - sin²(x) = cos²(x)
Substitute this into the expression
[tex]\frac{cos(x)(1 - sin(x))}{cos²(x)}[/tex]
[tex]\frac{cos(x)}{1+sin(x)}[/tex] = [tex]\frac{1 - sin(x)}{cos(x)}[/tex]
(first choice)
2) What is equivalent to 1 + cot⁴x?
Use this trigonometric identity
cos²(x) + sin²(x) = 1
Divide both sides by sin²(x)
cot²(x) + 1 = csc²(x)
1 = csc²(x) - cot²(x)
Multiply both sides by csc²(x) + cot²(x)
csc²(x) + cot²(x) = csc⁴(x) - cot⁴(x)
cot⁴(x) = csc⁴(x) - csc²(x) - cot²(x)
Add 1 to both sides
1 + cot⁴(x) = 1 + csc⁴(x) - csc²(x) - cot²(x)
Let's return to cot²(x) + 1 = csc²(x)
Multiply both sides by -1
-cot²(x) - 1 = -csc²(x)
-csc²(x) - cot²(x) = -2cot²(x) - 1
Substitute this into 1 + cot⁴(x) = 1 + csc⁴(x) - csc²(x) - cot²(x)
1 + cot⁴(x) = 1 + csc⁴(x) - 2cot²(x) - 1
1 + cot⁴(x) = csc⁴(x) - 2cot²(x)
(third choice)
