How do you simplify \[1 - {\cos ^2}\theta \]?

Hint: Here, we will use the values of trigonometric functions in a right-angled triangle to find the sine and cosine functions. Then we will square and add both the functions and apply Pythagoras Theorem to find the relationship between the sum of squares of sine and cosine. Using this we will be able to find the value of the given expression.

Complete step by step solution:
As we know, in a right angled triangle, if the lengths of the hypotenuse, base and perpendicular sides are \[H,B,P\] respectively, then,
Then the values of the trigonometric functions:
\[\sin \theta = \dfrac{P}{H}\]
And, \[\cos \theta = \dfrac{B}{H}\]
Now, squaring and adding both of them, we get,
\[{\sin ^2}\theta + {\cos ^2}\theta = \dfrac{{{P^2}}}{{{H^2}}} + \dfrac{{{B^2}}}{{{H^2}}}\]
Since, the denominator is the same, thus, we will simply add the numerators. Therefore, we get
\[ \Rightarrow {\sin ^2}\theta + {\cos ^2}\theta = \dfrac{{{P^2} + {B^2}}}{{{H^2}}}\]…………………………..\[\left( 1 \right)\]
Now according to Pythagoras Theorem, we know that in a right angled triangle, the sum of square of the two sides in a right angled triangle are equal to the square of the longest side i.e. the hypotenuse.
Hence, we get that \[{P^2} + {B^2} = {H^2}\]
Substituting this in the numerator of equation \[\left( 1 \right)\], we get
\[ \Rightarrow {\sin ^2}\theta + {\cos ^2}\theta = \dfrac{{{H^2}}}{{{H^2}}} = 1\]
Now, substituting the value of 1 in \[1 - {\cos ^2}\theta \], we get
\[ \Rightarrow 1 - {\cos ^2}\theta = {\sin ^2}\theta + {\cos ^2}\theta - {\cos ^2}\theta \]

\[ \Rightarrow 1 - {\cos ^2}\theta = {\sin ^2}\theta \]

Note:
In this question, we have used trigonometry. Trigonometry is a branch of mathematics that helps us to study the relationship between the sides and the angles of a triangle. In practical life, trigonometry is used by cartographers (to make maps). It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine, and the cosine function. In simple terms, they are written as ‘sin’, ‘cos’, and ‘tan’. Hence, trigonometry is not just a chapter to study, in fact, it is being used in everyday life.