To simplify the expression $\sqrt{(1 - \sin^2\theta) \div (1 - \cos^2\theta)}$, we use fundamental trigonometric identities.
Simplification Steps
- Recall the Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$.
- From this identity, derive two useful forms:
- $1 - \sin^2\theta = \cos^2\theta$
- $1 - \cos^2\theta = \sin^2\theta$
- Substitute these into the expression under the square root: $ \sqrt{\frac{\cos^2\theta}{\sin^2\theta}} $
- Simplify the fraction inside the square root: $ \sqrt{\left(\frac{\cos\theta}{\sin\theta}\right)^2} $
- Simplify the square root. Assuming $\frac{\cos\theta}{\sin\theta}$ is non-negative or considering the principal value: $ \frac{\cos\theta}{\sin\theta} $
- Recognize the final trigonometric ratio: $ \cot\theta $
Final Result
The simplified form of the expression $\sqrt{(1 - \sin^2\theta) \div (1 - \cos^2\theta)}$ is $\cot\theta$.
Similar Questions
- If $\cos(x + y) = \frac{1}{2}$ and $\sin(x - y) = 0$, where x and y are positive acute angles and $x \ge y$, then x and y are:
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- If $\tan A = \frac{3}{4}$, then the value of $\frac{\cos A - \sin A}{\cos A + \sin A}$ is:
- What is the value of $\sin(48^\circ + \theta) - \cos(42^\circ - \theta)$?
- What is the value of the following expression?
$(\tan 2^\circ \tan 88^\circ) (\tan 3^\circ \tan 87^\circ) ... (\tan 43^\circ \tan 47^\circ) \tan 45^\circ$ - If the ratio of the sine of an acute angle to its cosine is 5 : 12, then what will the value of sine of that angle be?
- If $\tan\theta + \cot\theta = 6$, then find the value of $\tan^2\theta + \cot^2\theta$
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$\frac{\cos 3x + \cos x}{\sin 3x - \sin x}$ - If $\sin(A - B) = \frac{1}{2}$ and $\cos(A + B) = \frac{1}{2}$ with $0^\circ < (A + B) \leq 90^\circ$, $A > B$, then find the measure of A and B.
- If $\sqrt{2} \sin(5x - 5)^\circ = \tan 45^\circ$, then the value of x (in degrees) is:
Important Questions from Trigonometric Ratios and Identities
What is the ratio of the greatest to the smallest value of 2 – 2 sin x – sin 2x, 0 ≤ x ≤ (π/2)?
If sinθ = \(\frac{4}{5}\) , Find the value of sin3θ
If x, y are acute angles, where 0 < x + y < 90° and sin(3x - 40°) = cos (3y + 40°), then the value of tan (x + y) is equal to
What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?
If \(\sin \theta =\frac{3}{5}\) and \(\cos \theta =\frac{4}{5}\) , then the value of \(\frac{1+\tan \theta}{1-\cot \theta}\) is:
