Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\(\text{(sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A}\)
Solution and Explanation
\(\text{(sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A}\)
L.H.S =\(\text{(sin A + cosec A)² + (cos A + sec A)²}\)
\(⇒\text{ sin² A + cosec² A + 2sin A cosec A + cos² A + sec² A + 2cos A sec A}\)
Using the reciprocal identities \(\text{sec A} = \frac{1}{\text{cos A}}\) and \(\text{cosec A} =\frac{ 1}{\text{sin A}}\), and rearranging terms:
\(⇒\text{ (sin² A + cos² A) + (cosec² A + sec² A)+ 2 sin A }(\frac{1}{\text{sin A}}) +\text{2cos A }(\frac{1}{\text{cos A}})\)
\(⇒ \text{1 + 1 + cot² A + 1 + tan² A + 2 + 2}\)
\(= \text{7 + tan² A + cot² A}\)
= R.H.S
Top Questions on Trigonometric Identities
If \[ \frac{\cos^2 48^\circ - \sin^2 12^\circ}{\sin^2 24^\circ - \sin^2 6^\circ} = \frac{\alpha + \beta\sqrt{5}}{2}, \] where \( \alpha, \beta \in \mathbb{N} \), then the value of \( \alpha + \beta \) is ___________.
- JEE Main - 2026
- Mathematics
- Trigonometric Identities
If $\cot x=\dfrac{5}{12}$ for some $x\in(\pi,\tfrac{3\pi}{2})$, then \[ \sin 7x\left(\cos \frac{13x}{2}+\sin \frac{13x}{2}\right) +\cos 7x\left(\cos \frac{13x}{2}-\sin \frac{13x}{2}\right) \] is equal to
- JEE Main - 2026
- Mathematics
- Trigonometric Identities
The value of \(\dfrac{\sqrt{3}\cosec 20^\circ - \sec 20^\circ}{\cos 20^\circ \cos 40^\circ \cos 60^\circ \cos 80^\circ}\) is equal to
- JEE Main - 2026
- Mathematics
- Trigonometric Identities
- Prove the identity \( \sec^2 \theta = 1 + \tan^2 \theta \) for any right-angled triangle and use it to show that: \[ \frac{\sin \theta - \cos \theta + 1}{\sin \theta + \cos \theta - 1} = \frac{1}{\sec \theta - \tan \theta}. \]
- UK Class X - 2026
- Mathematics
- Trigonometric Identities
- Want to practice more? Try solving extra ecology questions todayView All Questions
