Question

Show that d= 2v² costheta sintheta(0_x)/gcos²thetax

Answer 1

Since the question is unclear, we assume as follows:

Show that $\dfrac{\cos \theta-\sin \theta}{\cos \theta+\sin \theta}=\sec 2 \theta-\tan 2 \theta$ .

Solution:

$\dfrac{\cos \theta-\sin \theta}{\cos \theta+\sin \theta}$

multiplying and dividing by \cos \theta-\sin \theta

$=\frac{\cos \theta-\sin \theta}{\cos \theta+\sin \theta} \times \frac{\cos \theta-\sin \theta}{\cos \theta-\sin \theta}$

simplifying and using trigonometric identities

$\begin{aligned} &=\frac{(\cos \theta-\sin \theta)^{2}}{\cos ^{2} \theta-\sin ^{2} \theta}=\frac{\cos ^{2} \theta+\sin ^{2} \theta-2 \sin \theta \cos \theta}{\cos 2 \theta} \\ &=\frac{1-\sin 2 \theta}{\cos 2 \theta}=\sec 2 \theta-\tan 2 \theta \end{aligned}$