Question

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

• A. $-1$
• B. $0$
• C. $+1$
• D. $+2$

Answer 1

Answer: (D)

$+2$

$(1-\tan A+\text{sec}A)(1-\cot A+\text{cosec}A)=(1-\frac{\sin A}{\cos A}+\frac{1}{\cos A})(1-\frac{\cos A}{\sin A}+\frac{1}{\sin A})$

$=\frac{(1+\cos A-\sin A)(1+\sin A-\cos A)}{\sin A\cos A}$

$=\frac{(1+\cos A-\sin A)(1-(\cos A-\sin A\left)\right)}{\sin A\cos A}$

$=\frac{1-(\cos A-\sin A{)}^2}{\sin A\cos A}$

$=\frac{1-({\sin }^{2}A+{\cos }^{2}A-2\sin A\cos A)}{\sin A\cos A}$

$=\frac{1-1+2\sin A\cos A}{\sin A\cos A}$

$=+2$