Question

Question #cf633

Answer 1

Prove: `1/(1+cos(x))=csc(x)(csc(x)-cot(x))`

Substitute `csc(x) = 1/sin(x)` and `cot(x) = cos(x)/sin(x)`:

`1/(1+cos(x))=1/sin(x)(1/sin(x)-cos(x)/sin(x))`

Combine the two fractions:

`1/(1+cos(x))=1/sin(x)((1-cos(x))/sin(x))`

Perform the multiplication:

`1/(1+cos(x))=(1-cos(x))/sin^2(x)`

Substitute `sin^2(x) = 1-cos^2(x)`:

`1/(1+cos(x))=(1-cos(x))/(1-cos^2(x))`

Factor the denominator:

`1/(1+cos(x))=(1-cos(x))/((1-cos(x))(1+cos(x)))`

The common factor in the numerator and denominator becomes 1:

`1/(1+cos(x))=1/(1+cos(x))` Q.E.D.