Question 592883
These identities can be hard because there is no "recipe" that one can memorize and then apply to each one.<br>
When I looked at your problem here is what I saw:<ul><li>The right side is expressed in terms of sec(x)</li><li>Since sec(x) is the reciprocal of cos(x) it might be helpful to find a way to express the left side in terms of cos(x)</li><li>The left side is expressed in terms of sin(x). One way to convert sin's to cos's is to use sin^2(x) + cos^2(x) = 1 or cos^2(x) = 1 - sin^2(x)</li><li>The sin's on the left are not squared. But from the factoring pattern {{{(a+b)(a-b) = a^2 - b^2}}} I know I can get 1 - sin^2(x) in both denominators by multiplying each fraction by appropriate expressions (as you'll see shortly). This will turn both denominators into cos^2(x)! And not only that, since the denominators will then be the same, the fractions can then be added!!</li><li>Adding the fractions will turn the left side into a single term (which is good because the right side is also a single term).</li></ul>
So let's put these ideas into action:
{{{1/(1-sin(x))+1/(1+sin(x))=2sec^2(x)}}}
Create the {{{1-sin^2(x)}}} denominators:
{{{((1+sin(x))/(1+sin(x)))*(1/(1-sin(x)))+((1-sin(x))/(1-sin(x)))(1/(1+sin(x)))=2sec^2(x)}}}
which simplifies to:
{{{(1+sin(x))/(1-sin^2(x))+(1-sin(x))/(1-sin^2(x))=2sec^2(x)}}}
The denominators can be replaced by cos^2(x):
{{{(1+sin(x))/cos^2(x)+(1-sin(x))/cos^2(x)=2sec^2(x)}}}
And we can add the fractions together. The sin's in the numerator cancel out giving us:
{{{2/cos^2(x)=2sec^2(x)}}}
We are very close now. Factoring out 2 on the left (after all, 2 is a factor on the right):
{{{2(1/cos^2(x))=2sec^2(x)}}}
And the fraction on the left can be replaced by sec^2(x):
{{{2sec^2(x)=2sec^2(x)}}}
And we're done!