Write 'True' or 'False' and justify your answer.
The value of the expression $(\cos^{2} 23^{\circ} - \sin^{2} 67^{\circ})$ is positive.

Given that $\alpha + \beta = 90^{\circ}$,show that $\sqrt{\cos \alpha \operatorname{cosec} \beta - \cos \alpha \sin \beta} = \sin \alpha$.

If $\sec 4A = \operatorname{cosec}(A - 20^{\circ})$,where $4A$ is an acute angle,find the value of $A$ (in $^{\circ}$).

Write 'True' or 'False' and justify your answer.
$(\tan \theta+2)(2 \tan \theta+1)=5 \tan \theta+\sec ^{2} \theta$

The value of the expression $\left[\frac{\sin ^{2} 22^{\circ}+\sin ^{2} 68^{\circ}}{\cos ^{2} 22^{\circ}+\cos ^{2} 68^{\circ}}+\sin ^{2} 63^{\circ}+\cos 63^{\circ} \sin 27^{\circ}\right]$ is:

If $a \sin \theta + b \cos \theta = c$,then prove that $a \cos \theta - b \sin \theta = \pm \sqrt{a^2 + b^2 - c^2}$,given $a^2 + b^2 \geq c^2$.