Prove that cos x/(1-sin x)=tan (π/4+x/2).

Question

Prove that [latex] \frac{\cos x}{(1-\sin x)}= an \left(\frac{\pi}{4}+\frac{x}{2} ight) [/latex].

Accepted Answer

We have

[latex]\begin{array}{l} \text { LHS }=\frac{\cos x}{(1-\sin x)} \ \=\frac{\left(\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}\right)}{\left(\cos ^{2} \frac{x}{2}+\sin ^{2} \frac{x}{2}-2 \sin \frac{x}{2} \cos \frac{x}{2}\right)} \ \\left[ \begin{array}{l} \because \cos x=\left(\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}\right), \cos ^{2} \frac{x}{2}+\sin ^{2} \frac{x}{2}=1\ \text{ and } \sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2} \end{array} \right] \end{array} \ \[/latex]

[latex]\begin{array}{l} =\frac{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)}{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)^{2} }\ \=\frac{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)}{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)}=\frac{\left(1+\tan \frac{x}{2}\right)}{\left(1-\tan \frac{x}{2}\right)}=\frac{\left(\tan \frac{\pi}{4}+\tan \frac{x}{2}\right)}{\left(1-\tan \frac{\pi}{4} \cdot \tan \frac{x}{2}\right)} \ \ [ \text{ dividing num. and denom. by cos} \frac{x}{2} ]\ \=\tan \left(\frac{\pi}{4}+\frac{x}{2}\right)=\text { RHS. }\end{array}[/latex]

Question 15.5.5: Prove that cos x/(1-sin x)=tan (π/4+x/2)....

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