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Trigonometric Identities

Trigonometric Identities

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Trigonometric identities are important to work through a variety of problems and advanced equations. They allow us to simplify many problems and make situations easier.

What is the main set of trigonometric identities?

There are two main formulaic identities that must be learnt to prove and solve other equations. These are:

and sin2xcos2x=tanx

Let’s prove these identities starting with sin2x+cos2x=1.


Firstly let’s draw a triangle with angle θ.

Trigonometric identities Triangle StudySmarter

General Triangle of angle θ

Now if we write out expressions for a and b using SOHCAHTOA we get:a=csinθb=ccosθ



Now if we square both of these expressions for sin and cos we get:


Summing these we get:


By Pythagoras' theorem:




Now let’s move on to proving sinxcosx=tanx. The first half of this proof is identical to the proof above.


Firstly let’s draw a triangle with angle θ.

Trigonometric identities Triangle StudySmarterNow if we write out expressions for a and b using SOHCAHTOA we get:So Now if we divide these two expressions for sin and cos:sinθcosθ=(ac)(bc)=ac×cb=ab

This is an expression for the opposite side over the adjacent side, therefore:




Now let’s look at some worked examples where trigonometric identities can be applied.

Worked examples using trigonometric identities

Solve the equation 4sin2x+8cosx-7=0 for 0x180.

SOLUTION:The first thing to do would be to substitute1-cos2x for sin2x .The equation now ends up being 4(1-cos2x)+8cosx-7=0 .Simplifying this further:4-4cos2x+8cosx-7=04cos2x-8cosx+3=0Now we can solve this like a quadratic by taking y=cosx.4y2-8y+3=0(2y-1)(2y-3)=0y=0.5 or y=1.5Now we need to do x = cos-1(y)We can only perform cos-1(0.5)=60°This is because 1.5 > 1 so we cannot perform a cos-1 function of this.So the only answer is 60°.

Let's look at another example of rearranging trigonometric identities.

Show that the equation 2sinx=(4cosx-1)tanx can be written as 6cos2x - cosx - 2 = 0.

SOLUTION:Firstly let’s rearrange to get rid of any denominators.2sinxtanx=4cosx-1Now let’s replace tanx with sinxcosx:2sinxsinxcosx=4cosx-12sin2xcosx=4cosx-1Now get rid of the denominator by multiplying through by cosx:2sin2x=4cos2x-cosxNow replace sin2x with 1-cos2x:2(1-cos2x)=4cos2x-cosx2-2cos2x=4cos2x-cosxNow rearrange this equation:2=6cos2x-cosx6cos2x-cosx-2 = 0QED

What other trigonometric identities can we derive?

Firstly we need to know three new bits of terminology:


These are all reciprocals of standard sin, cos and tan.

Deriving new identities

Now let’s look at the identity sin2x+cos2x=1:

If we divide the entire equation by cos2(x)we get:sin2xcos2x+cos2xcos2x=1cos2xNow using the identity sinxcosx=tanx:tan2x+1=sec2xThis is our first new identity. Now if we divide our entire equation by sin2xsin2xsin2x+cos2xsin2x=1sin2xNow using the identity sinxcosx=tanx, so :1+1tan2x=1sin2x1+cot2x=cosec2xNow we have our two new identities:tan2x+1=sec2xcot2x+1=cosec2x

Let’s see them in action in some worked examples.

Worked examples of new identities

Solve, for 0 ≤ θ < 360°, the equation:

2tan2x+secx=1to 1 dp.

Graph of y=cosx. Image: Ruben Verhaegh, CC BY-SA 4.0

We can see that if we perform the identity cosx=cos(360-x), the other value of x is 360-131.8=228.2.

Then we need to perform cos-1(1)=0, again using the identity cosx=cos(360-x), x=360.

So to 1 decimal place our 4 solutions in degrees are:


Trigonometric Identities - Key takeaways

    • Trigonometric identities are used to derive new formulae and equations.

    • They can help solve equations involving trigonometry.

    • They help us geometrically visualise real-life situations.

    • They have proofs, which can be adapted from basic trigonometry.


Graph of y=cos x: https://commons.wikimedia.org/wiki/File:Cos(x).PNG

Frequently Asked Questions about Trigonometric Identities

sinx/cosx=tanx, sin^2(x)+cos^2(x)=1. 1/cosx=secx

Simply rearrange to the identities listed above and substitute them back in.

Drawing a diagram reveals why each identity works. Regular SOHCAHTOA can show what’s going on.

They can help us solve larger trigonometric equations that cannot be solved otherwise.

Final Trigonometric Identities Quiz

Trigonometric Identities Quiz - Teste dein Wissen


What theorem are Pythagorean identities based on?

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Pythagoras Theorem

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