M L Aggarwal Solutions for Chapter: Trigonometric Identities, Exercise 5: Chapter Test

Evaluate the following:

$2\left(\frac{{\cos }^2 20°+{\cos }^2 70°}{{\sin }^2 25°+{\sin }^2 65°}\right)-\tan 45°+\tan 13°\tan 23°\tan 30°\tan 67°\tan 77°$

$\frac{\sec 29°}{\mathrm{cosec} 61°}+2 \cot 8° \cot 17° \cot 45° \cot 73° \cot 82°-3\left({\sin }^2 38°+{\sin }^2 52°\right)$

$\frac{{\sin }^2 22°+{\sin }^2 68°}{{\cos }^2 22°+{\cos }^2 68°}+{\sin }^2 63°+\cos 63° \sin 27°$

$\mathrm{cosec}\left(75°+\mathrm{\theta }\right)-\sec \left(15°-\mathrm{\theta }\right)-\tan \left(55°+\mathrm{\theta }\right)+\cot \left(35°-\mathrm{\theta }\right)$

If $\frac{4}{3}({\sec }^2 59°-{\cot }^2 31°)-\frac{2}{3}\sin 90°+3{\tan }^{2 }56° {\tan }^{2 }34°=\frac{x}{2}$, then find the value of $x$.

If $\theta$ is an acute angle such that $\sec \mathrm{\theta }=\mathrm{cosec} 60°$, find the value of $2 {\cos }^2\theta -1$.

If $5\mathrm{A}$ is an acute angle such that $\sec 5\mathrm{A}=\mathrm{cosec}\left(\mathrm{A}+36°\right)$, find the value of $\mathrm{A}$.

If $A, B$ and $C$ are the interior angles of a triangle $ABC,$ show that, $\sin \frac{A+B}{2}=\cos \frac{C}{2}$