Question

How do you simplify the expression \[1 - {\sin ^2}\theta \] ?

Answer 1

Hint: In this question we need to find the simplified form of \[1 - {\sin ^2}\theta \]. The trigonometry is the part of calculus and the basic ratios of trigonometric are sine and cosine which have their application in sound and light wave theories. The trigonometric have vast applications in naval engineering such as to determine the height of the wave and the tide in the ocean.

Complete Step by Step solution:
In this question we have given the trigonometric expression as \[1 - {\sin ^2}\theta \] and we need to simplify the expression.
The given function is simplified by using the trigonometric identity.
We will consider the general trigonometric identity \[{\sin ^2}\theta + {\cos ^2}\theta = 1\]
Now, we write the above trigonometric identity in term of the cosine function as,
\[ \Rightarrow {\sin ^2}\theta - {\sin ^2}\theta + {\cos ^2}\theta = 1 - {\sin ^2}\theta \]
Subtract \[{\sin ^2}\theta \] on both the sides of the equation.
Then, we write the solution as,
\[ \Rightarrow {\cos ^2}\theta = 1 - {\sin ^2}\theta \]
Now, we will exchange the Left-hand side and the right-hand side of the expression as,
\[\therefore 1 - {\sin ^2}\theta = {\cos ^2}\theta \]
Thus, this gives the result.

From above the simplified form of \[1 - {\sin ^2}\theta \] is \[{\cos ^2}\theta \].

Note:
As we know that the sine angle formula is used to determine the ratio of perpendicular to height in a right-angle triangle. It is also used to determine the missing sides and the angles in other types of triangles.