If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Using trigonometric identities

Trigonometric identities like sin²θ+cos²θ=1 can be used to rewrite expressions in a different, more convenient way. For example, (1-sin²θ)(cos²θ) can be rewritten as (cos²θ)(cos²θ), and then as cos⁴θ. Created by Sal Khan.

## Want to join the conversation?

• how is tan squared less 1 = secant? Each question for this section uses this central calculation to simplify the calculations, but it makes no logical sense
We must simplify (tan^2 theta - 1) <<<< note the 1 within this argument, we're taking an angle, and deducting 1
Start by simplifying the tan^2 theta angle
tan^2 = sin^2+cos^2 = 1 << this we can agree on
the solutions tell us to divide both sides by cos^2.
so sin^2/cos^2 + cos^2/cos^2 = 1/cos^2 and 1/cos^2 is sec^2 << still following
then somehow it says therefore tan^2-1 = sec^2 so it replaces the entire first argument with sec^2, completely ignoring that 1 we were supposed to deduct from tan.
how is this possible? tan^2 is equal to sec^2 according to the calculations, they're just ignoring the one at the end of that original argument we're trying to simplify, like it wasn't there.

If sin^2 + cos^2 =tan^2 = 1
then tan^2 - 1 should theoretically be 0, I know this isn't the answer, but you can see that the 1 in tan^2 - 1 can't be ignored, it's not the 1 from the calculation of tan^2, so how can the simplification of tan^2 wipe out this 1? • tan²θ = sin²θ + cos²θ = 1
That is wrong. tan²θ = sin²θ/cos²θ. Secondly, the identity is tan²θ + 1 = sec²θ, not tan²θ - 1.
Maybe this proof will be easier to follow:
tan²θ + 1
= sin²θ/cos²θ + 1
= sin²θ/cos²θ + cos²θ/cos²θ
= (sin²θ + cos²θ)/cos²θ //sin²θ + cos²θ = 1, which we substitute in.
= 1/cos²θ
= sec²θ
Therefore, tan²θ + 1 = sec²θ.
• • The process is somewhat confusing to find the exact value, but here it is:
Let x = 18° (therefore 5x = 90°)
sin(3x) = cos(90° - 3x) = cos(5x - 3x) = cos(2x)
sin(3x) = cos(2x) (Remember that x = 18°, so that is why this is true.)
3sin(x) - 4sin^3(x) = 1 - 2sin^2(x) (I expanded these.)

Let y = sin(x)
4y^3 - 2y^2 - 3y + 1 = 0
4y^3 - 2y^2 - 2y - (y - 1) = 0
(4y^3 - 2y^2 - 2y) - (y - 1) = 0
(y - 1)(4y^2 + 2y) - (y - 1) = 0
(y - 1)(4y^2 + 2y - 1) = 0
Remember that sin18° is a root, so you can use the quadratic equation to solve for it. Also remember that sin18° is positive, which will help you choose the right answer.

4y^2 + 2y - 1 = 0
y = (-2 ± √[4 - 4(4)(-1)]) / 2(4)
y = (-2 ± √20) / 8
y = (-2 ± 2√5) / 8
y = -1/4 ± √(5)/4
sin18° is positive, so it must be the positive root, not the negative root here. So, the exact value of sin18° is -1/4 + √(5)/4.
• • at sal adds cos^2 with sin^2 to get 1 i don't understand that • That is a very important identity that comes directly from applying the Pythagorean theorem on the unit circle.

In the video, he used the Pythagorean theorem to say x²+y² = 1, but in the graph, x = cos ⊝ and y = sin ⊝. Thus (cos ⊝)²+(sin ⊝)² = 1 and this is often written as cos² ⊝+ sin² ⊝ = 1.
• how they got Sin theta over cos theta 2 equals to tan 2 theta • How do we know which identity to use when finding the other answers. For example when do I use (2pi-x) or(pi-x), or even the negative version of those. I haven't found any examples explaining this. • I felt stumped by this too.

What helped me was practicing A LOT. I did all the problems in my text book, then compared my answers to the back of the book. Then I watched all the videos in this section at least a few times, and went back and did all the textbook problems again. At some point, it seemed easy instead of impossible and it felt like a miracle.

I think what happened is I got a feel for which identities to use just by working with them over and over again. Hope this helps!
• • Can someone help me with establishing an identity? I'm having a bit of trouble with those types of problems. • Basically, If you want to simplify trig equations you want to simplify into the simplest way possible. for example you can use the identities -
cos^2 x + sin^2 x = 1
sin x/cos x = tan x
You want to simplify an equation down so you can use one of the trig identities to simplify your answer even more.
some other identities (you will learn later) include -
cos x/sin x = cot x
1 + tan^2 x = sec^2 x
1 + cot^2 x = csc^2 x
hope this helped!  