Question

The expression $(1-\tan A+\sec A)(1-\cot A+\cos \sec A)$ has value

• A. -1
• B. 0
• C. 1
• D. 2

Answer 1

The correct option is D 2

$(1-\tan A+\sec A)(1-\cot A+cosecA)$
= $(1-\frac{\sin A}{\cos A}+\frac{1}{\cos A})(1+\frac{\cos A}{\sin A}+\frac{1}{\sin A})$]
= $\frac{(\cos A-\sin A+1)(\sin A-\cos A+1)}{\sin A\cos A}$
= $\frac{\cos A\sin A-{\cos }^{2}A+\cos A-{\sin }^{2}A+\sin A\cos A-\sin A+\sin A-\cos A+1}{\sin A\cos A}$
= $\frac{2\cos A\sin A+1-({\sin }^{2}A+{\cos }^{2}A)}{\sin A\cos A}$
= $2$ (Since, ${\sin }^{2}A+co{s}^{2}A=1$)