Proving Identities

DOCUMENT();

loadMacros('PGstandard.pl', 'PGML.pl', 'scaffold.pl', 'PGcourse.pl');

Context()->variables->are(t => 'Real');

# Redefine sin(x) to be e^(pi x).
Context()->functions->remove('sin');

package AltSin;
# The next line makes the function a function from reals to reals.
our @ISA = qw(Parser::Function::numeric);

sub sin {
    shift;
    my $x = shift;
    return CORE::exp($x * $pi);
}

package main;

# Add the new functions to the Context.
Context()->functions->add(sin => { class => 'AltSin', TeX => '\sin' });

BEGIN_PGML
This problem has three parts. A part may be open if it is correct or if it is
the first incorrect part. Clicking on the heading for a part toggles whether it
is displayed.

In this multi-part problem, we will use algebra to verify the identity

>>[`\displaystyle \frac{\sin(t)}{1 - \cos(t)} = \frac{1 + \cos(t)}{\sin(t)}.`]<<

END_PGML

Scaffold::Begin(is_open => 'correct_or_first_incorrect');

Section::Begin('Part 1');
BEGIN_PGML
First, the equation above can be rewritten as
[`\displaystyle \sin(t)
    = \left( \frac{1 + \cos(t)}{\sin(t)} \right) \cdot \Big(`]
[_]{'1 - cos(t)'}{15} [` \Big) `].
END_PGML
Section::End();

Section::Begin('Part 2');
BEGIN_PGML
Next, the equation can be rewritten as
[`\sin(t) \cdot \big(`] [_]{'sin(t)'}{15}
[`\big) = \big(1 + \cos(t)\big) \cdot \big(1 - \cos(t)\big)`].
END_PGML
Section::End();

Section::Begin('Part 3');
BEGIN_PGML
Finally, the equation can be rewritten as
[`\sin^2(t) =`] [_]{'1-(cos(t))^2'}{15}, which is true since
[`\cos^2(t) + \sin^2(t) = 1`].

Thus, the original identity can be derived by reversing these steps.
END_PGML
Section::End();

Scaffold::End();

ENDDOCUMENT();