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Find the value of \[{\sec ^2}\theta - {\tan ^2}\theta \].

Last updated date: 16th Jul 2024
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Hint: In the given question, we have to solve the trigonometric equation. We will use the trigonometric identities to solve the equation and arrive at the answer. We should know that \[{\sec ^2}\theta = \dfrac{1}{{{{\cos }^2}\theta }}\]and \[{\tan ^2}\theta = \dfrac{{{{\sin }^2}\theta }}{{{{\cos }^2}\theta }}\].

Complete step by step solution:
Trigonometry is one of the branches of mathematics that uses trigonometric ratios to find the angles and missing sides of a triangle. The trigonometric functions are the trigonometric ratios of a triangle. The trigonometric functions are sine, cosine, sectant, cosecant, tangent and cotangent.
The Trigonometric formulas or Identities are the equations which are valid in the case of Right-Angled Triangles. They are also called Pythagorean Identities.
We can solve the question as follows-
We know that \[\sec \theta = \dfrac{1}{{\cos \theta }}\]so we can say that \[{\sec ^2}\theta = \dfrac{1}{{{{\cos }^2}\theta }}\].
Similarly, \[\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\]so we can get \[{\tan ^2}\theta = \dfrac{{{{\sin }^2}\theta }}{{{{\cos }^2}\theta }}\].
Applying the above identities, we get,
\[ = \dfrac{1}{{{{\cos }^2}\theta }} - \dfrac{{{{\sin }^2}\theta }}{{{{\cos }^2}\theta }}\]
We have common denominator, so we can get,
\[ = \dfrac{{1 - {{\sin }^2}\theta }}{{{{\cos }^2}\theta }}\]
Now since \[{\sin ^2}\theta + {\cos ^2}\theta = 1\], we can get \[{\cos ^2}\theta = 1 - {\sin ^2}\theta \].
Applying the above identity, we get,
\[ = \dfrac{{{{\cos }^2}\theta }}{{{{\cos }^2}\theta }}\]
\[ = 1\]
Hence, \[{\sec ^2}\theta - {\tan ^2}\theta = 1\].

In the given case, we have converted the secant and tangent into sine and cosine so we can establish the relationship between the two variables and solve the question easily. The key to solve such a question is to identify which trigonometric identity will be useful and accordingly apply the same. We should generally convert the variables into sine and cosine because they are the basic identities.
There can be more than one way to solve the question, for example we can solve the given question by applying different identity as follows:
\[{\sec ^2}\theta = 1 + {\tan ^2}\theta \]
Substituting the above value in the given question, we get,
\[ = 1 + {\tan ^2}\theta - {\tan ^2}\theta \]
\[ = 1\].