List of top Mathematics Questions on Trigonometric Identities

Match List-I (Inverse Trigonometric function Principal values) with List-II:
The value of \( \cos\left(\frac{\pi}{6}-\cos^{-1}\left(-\frac{\sqrt3}{2}\right)\right) \) is equal to:
The value of \( \cot^{-1}\left[2\cos\left(2\sin^{-1}\frac{1}{2}\right)\right] \) is:
The principal value of \( \tan^{-1}(1) \) is:
The value of $\tan 48^\circ \tan 16^\circ \tan 42^\circ \tan 74^\circ$ is _____.
If \( \sec \theta + \tan \theta = p \), then \( \sec \theta - \tan \theta = \dots \) “‘latex id="n7o3ls"
If \( 0^\circ < A < 90^\circ \), then simplify \( \sqrt{\frac{1+\sin A}{1-\sin A}} \) “‘latex id="76r4p8"
Prove that : $\frac{2 \cos^3} \theta - \cos \theta{\sin \theta - 2 \sin^3 \theta} = \cot \theta$
The value of Tan A + Cot A will be :
If \( \tan(A + B) = \sqrt{3} \) and \( \tan(A - B) = \frac{1}{\sqrt{3}} \), find the values of \( A \) and \( B \).
A 1.5 m tall boy is standing at some distance from a 30 m tall building; find the distance he walked towards the building if the angle of elevation of the top increases from \(30^\circ\) to \(60^\circ\).
Prove the identity: \(\displaystyle \frac{\cos A}{1 + \sin A} + \frac{1 + \sin A}{\cos A} = 2 \sec A.\)

The value of $\dfrac{1+\cot^2 \theta}{1+\tan^2 \theta}$ will be:
 

Prove that \(\displaystyle \frac{\sin A-\cos A+1}{\sin A+\cos A-1}=\frac{1}{\sec A-\tan A}\).

Prove that $\dfrac{1+\sec\theta}{\sec\theta}=\dfrac{\sin^2\theta}{1-\cos\theta}$.
 

Prove that \(\cot^{-1}\left(\frac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}\right) = \frac{x}{2}\), \(x \in (0, \pi/4)\).
The value of $(\sec A + \tan A)(1 - \sin A)$ will be:

If $3 \cot A = 4$, then the value of $\dfrac{1 - \tan^2 A}{1 + \tan^2 A}$ will be:
 

Prove that \(\displaystyle \frac{\sin\theta-\cos\theta+1}{\sin\theta+\cos\theta-1}=\sec\theta+\tan\theta\).
Prove that \[ \sqrt{\frac{1-\cos\theta}{1+\cos\theta}} = \csc\theta - \cot\theta \]
Prove that \( \sin^{-1} x = \cos^{-1} \sqrt{1 - x^2} \).
Find the principal value of \( \text{cosec}^{-1}(-\sqrt{2}) \).
Find the principal value of \(\cos^{-1}\left(-\frac{1}{\sqrt{2}}\right)\).
If \( (\cos x)^y = (\sin y)^x \) then \( \frac{dy}{dx} \) is:
Find the principal value of \( \tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2}\right) + \sin^{-1}\left(-\frac{1}{\sqrt{2}}\right) \).