What is \(\sqrt {\frac{{\sec x - \tan x}}{{\sec x + \tan x}}} \) equal to?
Simplifying Trigonometric Expressions: Square Root of Secant and Tangent
We need to simplify the given trigonometric expression: \( \sqrt{\frac{{\sec x - \tan x}}{{\sec x + \tan x}}} \).
To simplify expressions involving the square root of a fraction with trigonometric terms, especially those involving secant and tangent, it is often helpful to multiply the numerator and the denominator inside the square root by the conjugate of either the numerator or the denominator. In this case, let's multiply the numerator and denominator by the conjugate of the numerator, which is \((\sec x + \tan x)\).
The expression becomes:
$\( \sqrt{\frac{{\sec x - \tan x}}{{\sec x + \tan x}}} = \sqrt{\frac{{(\sec x - \tan x)(\sec x + \tan x)}}{{{(\sec x + \tan x)(\sec x + \tan x)}}}} $\)
Now, let's simplify the numerator and the denominator separately inside the square root.
The numerator is of the form \((a-b)(a+b)\), which expands to \(a^2 - b^2\). Here, \(a = \sec x\) and \(b = \tan x\). So, the numerator becomes:
$\( (\sec x - \tan x)(\sec x + \tan x) = \sec^2 x - \tan^2 x $\)
We use the fundamental trigonometric identity:
$\( \sec^2 x - \tan^2 x = 1 $\)
Thus, the numerator simplifies to \(1\).
The denominator becomes:
$\( (\sec x + \tan x)(\sec x + \tan x) = (\sec x + \tan x)^2 $\)
Substituting these back into the expression inside the square root, we get:
$\( \sqrt{\frac{{\sec^2 x - \tan^2 x}}{{(\sec x + \tan x)^2}}} = \sqrt{\frac{{1}}{{(\sec x + \tan x)^2}}} $\)
Now we can take the square root of the numerator and the denominator:
$\( \sqrt{\frac{{1}}{{(\sec x + \tan x)^2}}} = \frac{{\sqrt{1}}}{{\sqrt{{(\sec x + \tan x)^2}}}} $\)
$\( = \frac{{1}}{{|\sec x + \tan x|}} $\)
Assuming that \( \sec x + \tan x \) is positive in the relevant domain, or considering the magnitude of the expression, this simplifies further to:
$\( \frac{{1}}{{\sec x + \tan x}} $\)
This result matches one of the given options.
Therefore, \( \sqrt{\frac{{\sec x - \tan x}}{{\sec x + \tan x}}} \) is equal to \( \frac{{1}}{{\sec x + \tan x}} \).
Revision Table: Key Trigonometric Identities
| Identity | Description |
|---|---|
| \( \sin^2 \theta + \cos^2 \theta = 1 \) | Pythagorean Identity |
| \( \sec^2 \theta - \tan^2 \theta = 1 \) | Pythagorean Identity derived from dividing the main identity by \( \cos^2 \theta \) |
| \( \csc^2 \theta - \cot^2 \theta = 1 \) | Pythagorean Identity derived from dividing the main identity by \( \sin^2 \theta \) |
| \( \sec \theta = \frac{1}{{\cos \theta}} \) | Reciprocal Identity |
| \( \tan \theta = \frac{{\sin \theta}}{{\cos \theta}} \) | Quotient Identity |
Additional Information on Trigonometric Simplification
Simplifying trigonometric expressions is a key skill in trigonometry and calculus. The process often involves using fundamental identities to rewrite the expression in a simpler form. Common techniques include:
- Converting all terms to sine and cosine.
- Using Pythagorean identities (\( \sin^2 x + \cos^2 x = 1 \), \( \sec^2 x - \tan^2 x = 1 \), \( \csc^2 x - \cot^2 x = 1 \)).
- Factoring algebraic forms (like difference of squares, perfect squares).
- Finding a common denominator for fractional expressions.
- Multiplying the numerator and denominator by the conjugate, especially when dealing with terms like \( (1 \pm \sin x) \), \( (\sec x \pm \tan x) \), etc.
- Using sum-to-product or product-to-sum identities (for more complex expressions).
Always look for opportunities to apply identities or algebraic manipulations to reduce the expression. Pay attention to potential domain restrictions based on the original expression (e.g., denominators cannot be zero).
Similar Questions
What is \(\frac{{\cos \theta }}{{1 + \sin \theta }} + \frac{1}{{\cot \theta }}\) equal to?
What is the value of sin 26° + sin 212° + sin 218° + … + sin 284° + sin 290°?
What is (tan x + tan y)(1 – cot x cot y) + (cot x + cot y)(1 – tan x tan y) equal to?
If θ lies in the first quadrant and \(\cot \theta = \frac{{63}}{{16}}\) , then what is the value of (sin θ + cos θ)?
What is sin 4θ - cos 4θ equal to for any real number θ?
What is cot 1° cot 23° cot 45° cot 67° cot 89° equal to?
Consider the following statements:
1. (sec 2θ - 1) (1 - cosec 2θ) = 1
2. sin θ (1 + cos θ) -1 + (1 + cos θ) (sin θ) -1 = 2 cosec θ
Which of the above is/are correct?If sec x cosec x = 2, then what is the tan nx + cot nx equal to?
If sinθ = (m 2– n 2)/(m 2+ n 2) and 0 < θ < π/2, then what is the value of cosθ?
- What is \((\sec\theta - \tan\theta) - \frac{1-\sin\theta}{1+\sin\theta}\) equal to?
Important Questions from Trigonometric Identities
What is \(\rm \frac{1+tan^2\theta}{1+cot^2\theta}-\left(\frac{1-tan\theta}{1-cot\theta}\right)^2\) equal to?
If 3sin θ + 5cos θ = 5, then the value of 5sin θ - 3cos θ is equal to:
If angle C of a triangle ABC is a right angle where a, b and c are the sides opposite to the angles A, B and C respectively then what is tan A + tan B equal to?
If \(\sin \left( {A - B} \right) = \frac{1}{2}\) and \(\cos \left( {A + B} \right) = \frac{1}{2}\) , where A > B > 0° and A + B is an acute angle, then the value of A is:
In the equation
\(\rm\cos^{-1} \dfrac{1-a^2}{1 + a^2} - \cos^{-1} \dfrac{1-b^2}{1 + b^2} = 2 tan^{-1} x\) value of x is
