tan⁡θ+sec⁡θ−1tan⁡θ−sec⁡θ+1=

# $\frac{\mathrm{tan}\theta +\mathrm{sec}\theta -1}{\mathrm{tan}\theta -\mathrm{sec}\theta +1}=$

1. A

$\frac{1-\mathrm{sin}\theta }{\mathrm{cos}\theta }$

2. B

$\frac{1+\mathrm{sin}\theta }{\mathrm{cos}\theta }$

3. C

$\frac{-\mathrm{cos}\theta }{1+\mathrm{sin}\theta }$

4. D

$\frac{\mathrm{cos}\theta }{1+\mathrm{sin}\theta }$

Register to Get Free Mock Test and Study Material

+91

Verify OTP Code (required)

### Solution:

$\frac{\mathrm{tan}\theta +\mathrm{sec}\theta -1}{\mathrm{tan}\theta -\mathrm{sec}\theta +1}=\frac{\mathrm{sin}\theta +1-\mathrm{cos}\theta }{\mathrm{sin}\theta -1+\mathrm{cos}\theta }$

$\begin{array}{l}=\frac{2\mathrm{sin}\frac{\theta }{2}\mathrm{cos}\frac{\theta }{2}+2{\mathrm{sin}}^{2}\frac{\theta }{2}}{2\mathrm{sin}\frac{\theta }{2}\mathrm{cos}\frac{\theta }{2}-2{\mathrm{sin}}^{2}\frac{\theta }{2}}=\frac{\mathrm{cos}\frac{\theta }{2}+\mathrm{sin}\frac{\theta }{2}}{\mathrm{cos}\frac{\theta }{2}-\mathrm{sin}\frac{\theta }{2}}\\ =\frac{{\left(\mathrm{cos}\frac{\theta }{2}+\mathrm{sin}\frac{\theta }{2}\right)}^{2}}{{\mathrm{cos}}^{2}\frac{\theta }{2}-{\mathrm{sin}}^{2}\frac{\theta }{2}}=\frac{1+\mathrm{sin}\theta }{\mathrm{cos}\theta }=\frac{1-{\mathrm{sin}}^{2}\theta }{\mathrm{cos}\theta \left(1-\mathrm{sin}\theta \right)}=\frac{\mathrm{cos}\theta }{1-\mathrm{sin}\theta }\end{array}$

Register to Get Free Mock Test and Study Material

+91

Verify OTP Code (required)