Question

The value of ${{\tan }^{2}}\theta -{{\sec }^{2}}\theta $ is equal to

Answer 1

Hint: From the question given we have been asked to find the value of ${{\tan }^{2}}\theta -{{\sec }^{2}}\theta $. To solve this question, we have to write the tan and sec in terms of sin and cos. As we know that $\tan \theta =\dfrac{\sin \theta }{\cos \theta }$ and $\sec \theta =\dfrac{1}{\cos \theta }$. And also, we know the identity ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$. By using all these formulas, we will get the required answer.

Complete step by step solution:
From the question given we have to find the value of
$\Rightarrow {{\tan }^{2}}\theta -{{\sec }^{2}}\theta $
To solve this question, we have to write the tan and sec in terms of sin and cos.
As we know that the tan in terms of sin and cos is,
$\Rightarrow \tan \theta =\dfrac{\sin \theta }{\cos \theta }$
As we know that the sec in terms of cos is,
$\Rightarrow \sec \theta =\dfrac{1}{\cos \theta }$
Now we have to substitute this in the given equation,
Now by substituting these in the given equation we will get,
$\Rightarrow {{\tan }^{2}}\theta -{{\sec }^{2}}\theta ={{\left( \dfrac{\sin \theta }{\cos \theta } \right)}^{2}}-{{\left( \dfrac{1}{\cos \theta } \right)}^{2}}$
Now by further simplification we will get,
$\Rightarrow {{\tan }^{2}}\theta -{{\sec }^{2}}\theta =\dfrac{{{\sin }^{2}}\theta -1}{{{\cos }^{2}}\theta }$
As we know the trigonometric identity that is,
$\Rightarrow {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$
Rearranging this equation, we will get,
$\Rightarrow {{\sin }^{2}}\theta -1=-{{\cos }^{2}}\theta $
Now substitute this in the above equation we will get,
$\Rightarrow {{\tan }^{2}}\theta -{{\sec }^{2}}\theta =\dfrac{{{\sin }^{2}}\theta -1}{{{\cos }^{2}}\theta }$
$\Rightarrow {{\tan }^{2}}\theta -{{\sec }^{2}}\theta =\dfrac{-{{\cos }^{2}}\theta }{{{\cos }^{2}}\theta }$
 Now by further simplification we will get,
$\Rightarrow {{\tan }^{2}}\theta -{{\sec }^{2}}\theta =-1$
Therefore, the value of ${{\tan }^{2}}\theta -{{\sec }^{2}}\theta $ is equal to $-1$.

Note: Students should recall all the formulas of trigonometry before doing this problem, student should know some trigonometric identities like ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$, ${{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1$, $\cos e{{c}^{2}}\theta -{{\cot }^{2}}\theta =1$.