Question

If secx+tanx=p, obtain the values of secx, tanx and sinx in terms of p.

Answer 1

Dear student
$$\begin{gathered} \tan \theta +sec\theta =pI \\ Weknow \\ se{c}^2\theta -{\tan }^2\theta =1 \\ i,e \\ \left(\tan \theta +sec\theta \right)\left(-\tan \theta +sec\theta \right)=1 \\ -\tan \theta +sec\theta =\frac{1}{p}II \\ AddingIandIIweget \\ 2sec\theta =p+\frac{1}{p}III \\ i.e\mathit{s}\mathit{e}\mathit{c}\mathit{\theta }\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\left(\mathbf{p}\mathbf{+}\frac{\mathbf{1}}{\mathbf{p}}\right) \\ SubtractingII\hspace{0.17em}fromIweget \\ 2\tan \theta =p-\frac{1}{p}IV \\ \mathit{t}\mathit{a}\mathit{n}\mathit{\theta }\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\left(\mathbf{p}\mathbf{-}\frac{\mathbf{1}}{\mathbf{p}}\right) \\ DividingIV\hspace{0.17em}byIIIweget \\ \frac{\tan \theta }{sec\theta }=\frac{p-{\displaystyle \frac{1}{p}}}{p+{\displaystyle \frac{1}{p}}} \\ i.e \\ \frac{\sin \theta }{\cos \theta sec\theta }=\frac{p^2-1}{p^2+1} \\ \mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathbf{=}\frac{{\mathbf{p}}^{\mathbf{2}}\mathbf{-}\mathbf{1}}{{\mathbf{p}}^{\mathbf{2}}\mathbf{+}\mathbf{1}} \\ Regards \end{gathered}$$