Question:
Find the value of $(1 + \cot A – \csc A)(1 + \tan A + \sec A) – 3(\sin^2 A + \cos^2 A)$.
$(1 + \cot A – \csc A)(1 + \tan A + \sec A) – 3(\sin^2 A + \cos^2 A)$ का मान ज्ञात कीजिए।
Options:
- a) -1
- b) 1
- c) -2
- d) 2
- a) -1
- b) 1
- c) -2
- d) 2
correct answer : a)
Explanation:
Let us substitute $A = 45^\circ$ to simplify the expression:
$\cot 45^\circ = 1$, $\csc 45^\circ = \sqrt{2}$
$\tan 45^\circ = 1$, $\sec 45^\circ = \sqrt{2}$
$\sin^2 45^\circ + \cos^2 45^\circ = 1$
Substitute these values:
$(1 + 1 – \sqrt{2})(1 + 1 + \sqrt{2}) – 3(1)$
$= (2 – \sqrt{2})(2 + \sqrt{2}) – 3$
$= (2^2 – (\sqrt{2})^2) – 3$
$= (4 – 2) – 3 = 2 – 3 = -1$.
Hence, option (a) is the correct answer.
व्यंजक को सरल करने के लिए $A = 45^\circ$ प्रतिस्थापित करें:
$\cot 45^\circ = 1$, $\csc 45^\circ = \sqrt{2}$
$\tan 45^\circ = 1$, $\sec 45^\circ = \sqrt{2}$
$\sin^2 45^\circ + \cos^2 45^\circ = 1$
इन मानों को प्रतिस्थापित करें:
$(1 + 1 – \sqrt{2})(1 + 1 + \sqrt{2}) – 3(1)$
$= (2 – \sqrt{2})(2 + \sqrt{2}) – 3$
$= (2^2 – (\sqrt{2})^2) – 3$
$= (4 – 2) – 3 = 2 – 3 = -1$।
अतः, विकल्प (a) सही उत्तर है।