Applications of Pythagorean identities
The fundamental identity $\sin^2\theta+\cos^2\theta\equiv1$sin2θ+cos2θ≡1 is used in simplifying expressions and in establishing further trigonometric identities.
For example, we may wish to verify that $1+\tan^2\theta=\sec^2\theta$1+tan2θ=sec2θ, a formula that occurs in certain problems in calculus. A strategy for this is to manipulate one side of the equation by means of the fundamental identity until it can be seen to be the same as the other side. In this case we might write
| $LHS$LHS | $=$= | $1+\tan^2\theta$1+tan2θ |
| $=$= | $1+\frac{\sin^2\theta}{\cos^2\theta}$1+sin2θcos2θ | |
| $=$= | $\frac{\cos^2\theta}{\cos^2\theta}+\frac{\sin^2\theta}{\cos^2\theta}$cos2θcos2θ+sin2θcos2θ | |
| $=$= | $\frac{\cos^2\theta+\sin^2\theta}{\cos^2\theta}$cos2θ+sin2θcos2θ | |
| $=$= | $\frac{1}{\cos^2\theta}$1cos2θ | |
| $=$= | $\sec^2\theta$sec2θ | |
| $=$= |
$RHS$RHS |
Example 1
Simplify the expression $\left(\cos\beta+1\right)\left(\cos\beta-1\right)$(cosβ+1)(cosβ−1) and write it in terms of the sine function.
Expanding the brackets gives $\cos^2\beta-1$cos2β−1. But this is $-\left(1-\cos^2\beta\right)$−(1−cos2β) and hence it is equivalent to $-\sin^2\beta$−sin2β.
Example 2
Given that $x=4\sin\theta$x=4sinθ and $y=5\cos\theta$y=5cosθ, find a relation between $x$x and $y$y that does not involve the trigonometric functions.
In order to use the Pythagorean identity we need the squares of the trigonometric functions. So, we write $x^2=16\sin^2\theta$x2=16sin2θ and $y^2=25\cos^2\theta$y2=25cos2θ. Then, $\frac{x^2}{16}=\sin^2\theta$x216=sin2θ and $\frac{y^2}{25}=\cos^2\theta$y225=cos2θ. On adding these two equations we obtain $\frac{x^2}{16}+\frac{y^2}{25}=\sin^2\theta+\cos^2\theta$x216+y225=sin2θ+cos2θ, and finally
$\frac{x^2}{16}+\frac{y^2}{25}=1$x216+y225=1
This equation fixes the points $\left(x,y\right)$(x,y) that describe an ellipse in the coordinate plane. Evidently, the same ellipse can be specified by the two trigonometric functions in the parameter $\theta$θ.
More Worked Examples
QUESTION 1
Simplify the expression $\frac{1}{1-\sin x}\times\frac{1}{1+\sin x}$11−sinx×11+sinx.
QUESTION 2
Prove the identity $2\cos^2\left(\theta\right)-3=-1-2\sin^2\left(\theta\right)$2cos2(θ)−3=−1−2sin2(θ).
QUESTION 3
If $x=4\sin\theta$x=4sinθ and $y=3\cos\theta$y=3cosθ, form an equation relating $x$x and $y$y that does not involve $\sin\theta$sinθ or $\cos\theta$cosθ.