Question

How do you simplify $\dfrac{{si{n^4}x - co{s^4}x}}{{si{n^2}x - co{s^2}x}}$ ?

Answer 1

Hint: In this question, the trigonometry equation is given. This equation can be solved with basic algebraic formulas. We will use the algebraic formula of ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$. And we will also apply the basic Pythagorean trigonometry formula ${\sin ^2}x + {\cos ^2}x = 1$to solve this equation.

Complete step by step solution:
In this question, we want to solve the following equation.
$ \Rightarrow \dfrac{{si{n^4}x - co{s^4}x}}{{si{n^2}x - co{s^2}x}}$ ...(1)
First, let us solve the numerator of the equation.
The numerator is:
 $ \Rightarrow si{n^4}x - co{s^4}x$
That is equal to,
$ \Rightarrow si{n^4}x - co{s^4}x = {\left( {{{\sin }^2}x} \right)^2} - {\left( {co{s^2}x} \right)^2}$
Now, let us apply the algebraic formula ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$.
Here the value of ‘a’ is equal to ${\sin ^2}x$, and the value of ‘b’ is equal to ${\cos ^2}x$.
$ \Rightarrow si{n^4}x - co{s^4}x = \left( {{{\sin }^2}x - co{s^2}x} \right)\left( {{{\sin }^2}x + co{s^2}x} \right)$
Let us apply the basic trigonometry Pythagorean formula ${\sin ^2}x + {\cos ^2}x = 1$in the above equation.
Then we will get,
$ \Rightarrow si{n^4}x - co{s^4}x = \left( {{{\sin }^2}x - co{s^2}x} \right)\left( 1 \right)$
That is equal to,
$ \Rightarrow si{n^4}x - co{s^4}x = {\sin ^2}x - co{s^2}x$
Substitute the value of $si{n^4}x - co{s^4}x$ in the equation (1).
$ \Rightarrow \dfrac{{si{n^4}x - co{s^4}x}}{{si{n^2}x - co{s^2}x}} = \dfrac{{{{\sin }^2}x - co{s^2}x}}{{si{n^2}x - co{s^2}x}}$
Let us simplify the above equation. Here, the numerator and the denominator of the equation are the same. If the number is divided by the same number then the answer will be 1.
Therefore,
$ \Rightarrow \dfrac{{si{n^4}x - co{s^4}x}}{{si{n^2}x - co{s^2}x}} = 1$

Hence, the value of $\dfrac{{si{n^4}x - co{s^4}x}}{{si{n^2}x - co{s^2}x}}$ is equal to 1.

Note:
The Pythagorean trigonometric identity is also known as Pythagorean identity. It is an identity expressing the Pythagorean theorem in terms of trigonometric functions along with the sine and cosine functions.
The identity is based on the sine and the cosine functions are as below.
${\sin ^2}x + {\cos ^2}x = 1$
There are other two identities which are called Pythagorean trigonometric identity :
The trigonometric identity involving the tangent and the secant functions follows from the Pythagorean theorem.
${\tan ^2}x = 1 + {\sec ^2}x$
The trigonometric identity involving the cotangent and the cosecant functions follows from the Pythagorean theorem.
${\cot ^2}x = 1 + \cos e{c^2}x$