A) \[{{\sin }^{2}}A{{\cos }^{2}}A\]
B) \[{{\sec }^{2}}A\,\,\text{cose}{{\text{c}}^{2}}A\]
C) \[2{{\sec }^{2}}A\,\,\text{cose}{{\text{c}}^{2}}A\]
D) None of these
Correct Answer: C
Solution :
| \[{{(1-\tan A)}^{2}}+{{(1+\tan A)}^{2}}+{{(1-\cot A)}^{2}}+{{(1+\cot A)}^{2}}\]\[=1+{{\tan }^{2}}A-2\tan A+1{{\tan }^{2}}A\] |
| \[+\,\,2\tan A+1+{{\cot }^{2}}A-2\cot A+1{{\cot }^{2}}A+2\cot A\]\[=4+2\,\,({{\tan }^{2}}A+{{\cot }^{2}}A)\] |
| \[=(2+2{{\tan }^{2}}A)+(2+2{{\cot }^{2}}A)=2{{\sec }^{2}}A\]\[+2\text{cose}{{\text{c}}^{2}}A\]\[=2\left( \frac{1}{{{\cos }^{2}}A}+\frac{1}{{{\sin }^{2}}A} \right)=2\left( \frac{{{\sin }^{2}}A+{{\cos }^{2}}A}{{{\sin }^{2}}A\cdot {{\cos }^{2}}A} \right)\] |
| \[=\frac{2\cdot (1)}{{{\sin }^{2}}A\cdot {{\cos }^{2}}A}=2{{\sec }^{2}}A\cdot \text{cose}{{\text{c}}^{2}}A\] |
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