Question

How do you simplify the expression $1-{{\sec }^{2}}\theta$?

Answer 1

We first simplify the numerator and the denominator part of $1-{{\sec }^{2}}\theta$ separately. We use the identities ${{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1$. We also know that the terms $\sec \theta$ and $\cos \theta$ are inverse of each other which gives $\sec \theta \times \cos \theta =1$. For both parts we get the value of the simplification as equal to $-{{\tan }^{2}}\theta$.

Complete step-by-step solution:
We have been given a trigonometric expression of $1-{{\sec }^{2}}\theta$.
For the expression we have the identity of ${{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1$.
We subtract ${{\sec }^{2}}\theta$ from the both sides of the equation ${{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1$.
$\begin{aligned} & {{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1 \\\ & \Rightarrow {{\sec }^{2}}\theta -{{\tan }^{2}}\theta -{{\sec }^{2}}\theta =1-{{\sec }^{2}}\theta \\\ & \Rightarrow -{{\tan }^{2}}\theta =1-{{\sec }^{2}}\theta \\\ \end{aligned}$
We get $1-{{\sec }^{2}}\theta =-{{\tan }^{2}}\theta$.
Therefore, the simplified form of $1-{{\sec }^{2}}\theta$ is $-{{\tan }^{2}}\theta$.
We can also solve it using the inverse theorem.
We know that the terms $\sec \theta$ and $\cos \theta$ are inverse of each other.
So, $\sec \theta \times \cos \theta =1$ as $\sec \theta =\dfrac{1}{\cos \theta }$.
We multiply ${{\cos }^{2}}\theta$ to both the terms of $1-{{\sec }^{2}}\theta$. We also need to keep ${{\cos }^{2}}\theta$ divided in the denominator.
From the exponent theorem we get that for two numbers $a$ and $b$ we have ${{a}^{2}}\times {{b}^{2}}={{\left( ab \right)}^{2}}$.
Then we multiply ${{\cos }^{2}}\theta$ with ${{\sec }^{2}}\theta$ and get ${{\cos }^{2}}\theta \times {{\sec }^{2}}\theta$. The value of ${{\cos }^{2}}\theta \times {{\sec }^{2}}\theta$ can be replaced as 1 as $\cos \theta \sec \theta =1$.
So, ${{\cos }^{2}}\theta \left( 1-{{\sec }^{2}}\theta \right)={{\cos }^{2}}\theta -{{\cos }^{2}}\theta \times {{\sec }^{2}}\theta ={{\cos }^{2}}\theta -1$.
Now we use the theorem ${{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1$. This gives ${{\cos }^{2}}\theta -1=-{{\sin }^{2}}\theta$.
So, $1-{{\sec }^{2}}\theta =\dfrac{{{\cos }^{2}}\theta \left( 1-{{\sec }^{2}}\theta \right)}{{{\cos }^{2}}\theta }=\dfrac{-{{\sin }^{2}}\theta }{{{\cos }^{2}}\theta }=-{{\tan }^{2}}\theta$.

Note: We need to remember that the final terms are square terms. The identities ${{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1$ and $\cos \theta \sec \theta =1$ are valid for any value of $\theta$. The division of the fraction part only gives $-{{\tan }^{2}}\theta$ as the solution.