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RRB NTPC Graduate Tier 1 2025 Shift-1 📅 09 Jun, 2025

The value of $3 + tan^{2}\phi + cot^{2}\phi - sec^{2}\phi\ cosec^{2}\phi$ is:

A
1
B
2
C
-1
D
0
Result Summary
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APEDIA

RRB NTPC Graduate Tier 1
2025 • 09 Jun, 2025 • Shift-1
The value of $3 + tan^{2}\phi + cot^{2}\phi - sec^{2}\phi\ cosec^{2}\phi$ is:
Correct Answer
1
Trigonometric Identity Application: The expression involves a complex terminal term: $sec^{2}\phi\ cosec^{2}\phi$. We can strategically convert this specific pr......
Ref/Book : RRB NTPC Graduate Tier 1 2025
Ref. Subject Trigonometry
Chapter -
Ref. Topic Identities
Pg. No -
💡 Analysis & Explanation
Trigonometric Identity Application ▼
The expression involves a complex terminal term: $sec^{2}\phi\ cosec^{2}\phi$. We can strategically convert this specific product into an addition using fundamental identities: $sec^{2}\phi = 1 + tan^{2}\phi$ and $cosec^{2}\phi = 1 + cot^{2}\phi$.
Algebraic Expansion ▼
Multiplying these two expanded forms gives: $(1 + tan^{2}\phi)(1 + cot^{2}\phi) = 1 + cot^{2}\phi + tan^{2}\phi + (tan^{2}\phi * cot^{2}\phi)$. Since the product of tan and cot is exactly 1, this massive expansion elegantly reduces to $2 + tan^{2}\phi + cot^{2}\phi$.
Substituting Back into the Equation ▼
We substitute this reduced value back into the original mathematical expression: $3 + tan^{2}\phi + cot^{2}\phi - (2 + tan^{2}\phi + cot^{2}\phi)$.
Final Simplification ▼
Distributing the negative sign strictly cancels out the squared terms, leaving just the numerical difference: $3 - 2 = 1$.
Conclusion ▼
The entire complex expression simplifies uniformly to 1.
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