Question

How do you simplify $\dfrac{{{{\sec }^4}x - {{\tan }^4}x}}{{{{\sec }^2}x + {{\tan }^2}x}}$?

Answer 1

Hint: The given question is the trigonometric expression and in order to solve this solve we have to use the properties of trigonometric functions. We also need to know basic algebraic identities and rules of exponents to simplify any complicated expression. In order to solve this question, we’ll figure out the relation between $\tan \theta $ and$\sec \theta $. We’ll be using square law in order to find the relation. Square laws basically are trigonometric identities based on Pythagoras theorem, we can also use Pythagoras theorem to prove them.

Formula used: ${a^2} - {b^2} = (a + b)(a - b)$
${\tan ^2}\theta + 1 = {\sec ^2}\theta $
\[{({x^m})^n} = {x^{m \times n}}\]

Complete step by step solution:
We are given,
$\dfrac{{{{\sec }^4}x - {{\tan }^4}x}}{{{{\sec }^2}x + {{\tan }^2}x}}$
To simplify we need to observe the pattern to apply the formula,
The above equation can also be written as,
 $ \Rightarrow \dfrac{{{{({{\sec }^2}x)}^2}^{} - {{({{\tan }^2}x)}^2}}}{{{{\sec }^2}x + {{\tan }^2}x}}$
We obtained it using the Product Rule of exponents in reverse order,
\[{({x^m})^n} = {x^{m \times n}}\]
Here, we can apply the property in the numerator,
${a^2} - {b^2} = (a + b)(a - b)$
Where,
$
  a = {\sec ^2}x \\
  b = {\tan ^2}x \\
 $
$ \Rightarrow \dfrac{{({{\sec }^2}{x^{}} + {{\tan }^2}x)({{\sec }^2}{x^{}} - {{\tan }^2}x)}}{{{{\sec }^2}x + {{\tan }^2}x}}$
Now, we can cancel the like terms from numerator and denominator.
\[ \Rightarrow ({\sec ^2}{x^{}} - {\tan ^2}x)\]
Here, we can apply the property ${\tan ^2}\theta + 1 = {\sec ^2}\theta $
\[ \Rightarrow {\sec ^2}{x^{}} - {\tan ^2}x = 1\]
$ \Rightarrow 1$
This is the required answer.

Note: To simplify the expressions containing trigonometry, we need to memorize the properties associated with it. Trigonometric Ratios portray the relationship between measurement of angles and the length of the side of a triangle. It will make questions easier to solve. It is suggested that while solving the question of trigonometry we should carefully scrutinize the pattern of the given function, relating it with identities and then we should apply the formulas according to the identity which has been observed. Also, if the same number is multiplied or divided from numerator and denominator in a fraction the value remains the same.