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# If ${\sec ^2}\theta + {\tan ^2}\theta + 1 = 2$, then find the value of $\sec \left( { - \theta } \right)$:A. $- 2$B. $- \dfrac{1}{2}$C. $1$D. $\pm 1$  Answer Verified
Hint : Solve using trigonometric identities.

Given that: ${\sec ^2}\theta + {\tan ^2}\theta + 1 = 2$
Converting the above equation in the terms of Sin and Cos, we get

$\Rightarrow \dfrac{1}{{{{\cos }^2}\theta }} + \dfrac{{{{\sin }^2}\theta }}{{{{\cos }^2}\theta }} + 1 = 2{\text{ }}\left( {\because \tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}{\text{ and }}\sec \theta = \dfrac{1}{{\cos \theta }}} \right) \\ \Rightarrow \dfrac{{1 + {{\sin }^2}\theta + {{\cos }^2}\theta }}{{{{\cos }^2}\theta }} = 2 \\ \Rightarrow \dfrac{{1 + 1}}{{{{\cos }^2}\theta }} = 2{\text{ }}\left( {\because {{\sin }^2}\theta + {{\cos }^2}\theta = 1} \right) \\ \Rightarrow \dfrac{2}{{{{\cos }^2}\theta }} = 2 \\ \Rightarrow {\cos ^2}\theta = 1 \\ \Rightarrow \cos \theta = \pm 1{\text{ }} \ldots \, \ldots \left( 1 \right) \\$
We know that, $\cos \left( { - \theta } \right) = \cos \left( \theta \right)$
$\therefore \sec \left( { - \theta } \right) = \sec \left( \theta \right) = \dfrac{1}{{\cos \theta }}$
Put the value of $\cos \theta$ from equation $\left( 1 \right)$, we get
$\sec \left( { - \theta } \right) = \pm 1$

Note: In these types of problems, where there is no direct formula for the given trigonometric terms, one should always try to convert them to some trigonometric terms which have some relation using trigonometric relations and identities so as to make the problem easier to calculate.

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Trigonometric Identities  Use of If Then Statements in Mathematical Reasoning  CBSE Class 11 Maths Chapter 3 - Trigonometric Functions Formulas  CBSE Class 8 Maths Chapter 9 - Algebraic Expressions and Identities Formulas  Trigonometric Identities - Class 10  CBSE Class 11 Maths Chapter 1 - Sets Formulas  Sec 90  Sec 0  Sec 30  Sec 60  