1

Problem

If the combined value of A and B is \$π/2\$, find the value of \$(1-sin^2(A)) (1-cos^2(A)) \times (1 + cot^2(B)) (1+ tan^2(B))\$.

2

Step

The given expression

\$(1- sin^2(A)) (1 - cos^2(A)) \times (1 + cot^2(B)) (1+ tan^2(B))\$

3

Step

Combine value A and B is 90° so here A+ B = 90° then A = 90° - B

4

Formula

\$(1+ tan^2(B)) = sec^2(B) \$ and \$1+ cot^2(B) = cosec^2 (B)\$.

5

Step

So here plug the formulas in the given expression:

\$(1- sin^2(A)) (1-cos^2(A)) \times cosec^2(B) \times sec^2(B)\$

6

Hint

\$sin^2(A) = sin^2(90 - B) = cos^2B \$ and \$cos^2A = cos^2(90 - B) = sin^2B\$.

7

Step

So now plug the hint in the above expression:

\$(1 - cos^2(B)) (1 - sin^2(B)) \times cosec^2(B) \times sec^2(B)\$

8

Formula

\$(1- cos^2(B) = sin^2(B)) "and" (1- sin^2(B) = cos^2(B))\$.

9

Step

Now substitute these formulas into the expression:

\$sin^2(B) cos^2(B) \times 1/ (cos^2(B)) \times 1/ (sin^2(B))\$ = 1

10

Solution

So, the value of the given expression is 1.

11

Sumup

Please summarize Problem, Clue, Hint, Formula, Steps and Solution

Choices

12

Choice-A

The expression involves squares of trigonometric functions, and squaring doesn’t always result in 2

Wrong 2

14

Choice-B

The simplified expression should be in a positive value, not a negative value

Wrong -1

15

Choice-C

The expression simplifies to 1, rather than the previously stated -2, which was incorrect

Wrong -2

16

Choice-D

This is correct due to correct calculation by using the formula

Correct 1

17

Answer

Option

D

18

Sumup

Please summarize choices