Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved. They are fundamental tools in trigonometry and are used to simplify expressions, solve equations, and prove other trigonometric results. The three most important Pythagorean identities are:

These identities relate the squares of trigonometric functions to each other, stemming from the Pythagorean theorem applied to the unit circle.

Formulae

The core identities are:

$sin^2 A + cos^2 A = 1$
$1 + tan^2 A = sec^2 A$
$1 + cot^2 A = csc^2 A$

Examples

Example-1: Simplify the expression $sin^2(x) + cos^2(x) + tan^2(x)$.

Solution: Using the identity $sin^2(x) + cos^2(x) = 1$, the expression simplifies to $1 + tan^2(x)$. Then, using $1 + tan^2(x) = sec^2(x)$, the final simplified expression is $sec^2(x)$.

Example-2: Prove that $\frac{1}{1-sin(x)} + \frac{1}{1+sin(x)} = 2sec^2(x)$.

Solution: Combining the fractions on the left-hand side (LHS):

LHS = $\frac{(1+sin(x)) + (1-sin(x))}{(1-sin(x))(1+sin(x))}$

LHS = $\frac{2}{1-sin^2(x)}$

Using the identity $sin^2(x) + cos^2(x) = 1$, we can rewrite $1 – sin^2(x)$ as $cos^2(x)$.

LHS = $\frac{2}{cos^2(x)}$

Since $\frac{1}{cos(x)} = sec(x)$, then $\frac{1}{cos^2(x)} = sec^2(x)$.

LHS = $2sec^2(x)$ = RHS. Hence Proved.

Theorem with Proof

Theorem: Prove the identity $sin^2 A + cos^2 A = 1$.

Proof: Consider a right-angled triangle with angle $A$. Let the opposite side be $o$, the adjacent side be $a$, and the hypotenuse be $h$.

By definition, $sin A = \frac{o}{h}$ and $cos A = \frac{a}{h}$.

Therefore, $sin^2 A = (\frac{o}{h})^2 = \frac{o^2}{h^2}$ and $cos^2 A = (\frac{a}{h})^2 = \frac{a^2}{h^2}$.

Adding these two equations: $sin^2 A + cos^2 A = \frac{o^2}{h^2} + \frac{a^2}{h^2} = \frac{o^2 + a^2}{h^2}$.

By the Pythagorean theorem, $o^2 + a^2 = h^2$.

Substituting this into the previous equation: $sin^2 A + cos^2 A = \frac{h^2}{h^2} = 1$.

Therefore, $sin^2 A + cos^2 A = 1$. Q.E.D. (Quod Erat Demonstrandum – which was to be demonstrated)

Common mistakes by students

Forgetting to square the trigonometric functions. Students often write $sin A + cos A = 1$, which is incorrect. The squares are crucial.
Incorrectly manipulating the identities. For example, confusing $sin^2 A + cos^2 A = 1$ with $sin A + cos A = 1$.
Not recognizing when to apply the identities. They may not see how to use the identities to simplify or solve a given problem.
Using wrong angle or unit during calculation.

Real Life Application

These trigonometric identities are critical in various fields. For instance:

Physics: Used extensively in wave mechanics, optics (analyzing light), and calculating projectile motion.
Engineering: Essential for structural analysis (designing bridges, buildings), signal processing, and electrical circuit analysis.
Computer Graphics: Used in 3D rendering and animation for calculating transformations and lighting effects.
Navigation: Calculating distances and angles used in navigation and surveying.

Fun Fact

The Pythagorean identities are named after the Greek mathematician Pythagoras, though they’re related to the Pythagorean Theorem. These identities connect the sides of a right-angled triangle to its angles, forming the basis for many trigonometric relationships!