1. Fundamental Trigonometric Identities
by M. Bourne
Before we start to prove trigonometric identities, we see where the basic identities come from.
Recall the definitions of the reciprocal trigonometric functions, csc θ, sec θ and cot θ from the trigonometric functions chapter:
`csc theta=1/(sin theta)`
`sec theta=1/(cos theta)`
`cot theta=1/(tan theta)`
Now, consider the following diagram where the point (x, y) defines an angle θ at the origin, and the distance from the origin to the point is r units:
From the diagram, we can see that the ratios sin θ and cos θ are defined as:
Now, we use these results to find an important definition for tan θ:
`(sin theta)/(cos theta)=(y/r)/(x/r)=y/rxxr/x=y/x`
Now also `tan theta=y/x`, so we can conclude:
`tan theta=(sin theta)/(cos theta)`
Ratios based on Pythagoras' Theorem
Also, for the values in the above diagram, we can use Pythagoras' Theorem and obtain:
y2 + x2 = r2
Dividing through by r2 gives us:
so we obtain the important result:
`sin^2\ theta + cos^2\ theta = 1`
We now proceed to derive two other related formulas that can be used when proving trigonometric identities.
It is suggested that you remember how to find the identities, rather than try to memorise each one.
Dividing sin2θ + cos2θ = 1 through by cos2θ gives us:
`(sin^2 theta)/(cos^2 theta)+1=1/(cos^2\ theta)`
`tan^2 theta + 1 = sec^2 theta`
Dividing sin2θ + cos2θ = 1 through by sin2θ gives us:
`1+(cos^2 theta)/(sin^2 theta)=1/(sin^2 theta`
`1 + cot^2 theta = csc^2 theta`
Trigonometric Identities Summary
`tan theta=(sin theta)/(cos theta)`
`sin^2 theta+cos^2 theta=1`
`tan^2 theta+1=sec^2 theta`
`1+cot^2 theta=csc^2 theta`
Proving Trigonometric Identities
- Learn well the formulas given above (or at least, know how to find them quickly). The better you know the basic identities, the easier it will be to recognise what is going on in the problems.
- Work on the most complex side and simplify it so that it has the same form as the simplest side.
- Don't assume the identity to prove the identity. This means don't work on both sides of the equals side and try to meet in the middle. Start on one side and make it look like the other side.
- Many of these come out quite easily if you express everything on the most complex side in terms of sine and cosine only.
- In most examples where you see power 2 (that is, 2), it will involve using the identity sin2θ + cos2θ = 1 (or one of the other 2 formulas that we derived above).
Using these suggestions, you can simplify and prove expressions involving trigonometric identities.
`(tan y)/(sin y)=sec y`
We express everything in terms of `sin y` and `cos y` and then simplify.
("LHS" stands for "left-hand side" of the equation and "RHS" means "right-hand side".)
`"LHS"=tan\ yxx1/(sin y)`
`=(sin y)/(cos y)xx1/(sin y)`
`sin y + sin y\ cot^2y = csc y`
`sin x\ cos x\ tan x = 1 − cos^2x`
1. Prove that
`tan x + cot x = sec x\ csc x`
2. Prove that
`(1+cos x)/(sin x)=(sin x)/(1-cos x)`
In this problem, it is easier to start from the RHS. We need to perform a trick.
We multiply numerator and denominator of the fraction by the conjugate of the denominator. (We saw this idea in the sections Rationalizing the Denominator (in surds) and in dividing complex numbers.)
This will give us an expression in the denominator that we can simplify, using sin2θ + cos2θ = 1.
`"RHS" = (sin x)/(1-cos x)`
`=(sin x)/(1 - cos x)xx (1 + cos x)/(1 + cos x)`
`=((sin x)(1 + cos x))/(1 - cos^2 x)`
`=((sin x)(1 + cos x))/(sin^2 x)`
`=(1 + cos x)/(sin x)`
`"LHS" = sin y + sin y\ cot^2y`
`=sin y 1/(sin^2y)`
` = csc\ y`
`"LHS" = sin x\ cos x\ tan\ x `
`=sin x\ cos x(sin x)/(cos x)`
`=1-cos^2x` (since `sin^2x+cos^2x=1`)
`"LHS"=tan\ x + cot\ x`
`=(sin x)/(cos x) + (cos x)/(sin x)`
`=(sin^2x+cos^2x)/(cos x\ sin x)`
`=1/(cos x\ sin x)`
`=1/(cos x)xx1/(sin x)`
`=sec\ x\ csc\ x`
5. Analytical Trigonometry - 1 - www.mastermathmentor.com- Stu SchwartzUnit 5 – Analytical Trigonometry – ClassworkA) Verifying Trig Identities: Definitions to know: Equality: a statement that is always true. example: 2 = 2, 3 + 4 = 7, 62=36, 2 3+5( )=6+10. Equation: a statement that is conditionally true, depending on the value of a variable. example: 2x+3=11, x"1( )2=25, x3"2x2+5x"12=0, 2sin"=1.Identity: a statement that is always true no matter the value of the variable. example: 2x+3x=5x, 4x"3( )=4x"12, x"1( )2=x2"2x+1, 1x"1"1x+1=2x2"1. In the last example, it could be argued that this is not an identity, because it is not true for all values of the variable (xcannot be 1 or -1). However, when such statements are written, we assume the domain is taken into consideration although we don’t always write it. So a better definition of an identity is: a statement that is always true for all values of the variable within its domain. The 8 Fundamental Trigonometric Identities: Trig Identities proofs (assuming in standard position) Reciprocal Identitiescsc=1sinsec=1coscot=1tanQuotient Identitiestan=sincoscot=cossinPythagorean Identitiessin2+cos2=11+tan2=sec21+cot2=csc21sin=1yr=ry=csc1cos=1xr=rx=sec1tan=1yx=xy=cotsincos=yrxr=yx=tancossin=xryr=xy=cotx2+y2=r2x2+y2=r2x2+y2=r2x2r2+y2r2=r2r2x2x2+y2x2=r2x2x2y2+y2y2=r2y2cos2+sin2=1 | 1+tan2=sec2| cot2+1=csc2Corollaries: a statement that is true because another statement is true: Examples (you write the others): Reciprocal identities: sincsc=1 sin=1cscsincos=1Quotient identities: tancos=sincos=sintanPythagorean identities: sin2=1#cos2cos2=1#sin2sin= ±#cos2cos= ±#sin2