The value of 3 + tan² θ + cot² θ − sec² θ cosec² θ is:

Question

Question

This question was previously asked in

RRB NTPC Graduate Level CBT-I Official Paper (Held On: 09 Jun, 2025 Shift 1)

Answer 1

Given:

Expression: 3 + tan²θ + cot²θ − sec²θ × cosec²θ

Formula used:

tan²θ = sec²θ − 1

cot²θ = cosec²θ − 1

Calculation:

⇒ 3 + (sec²θ − 1) + (cosec²θ − 1) − sec²θ × cosec²θ

⇒ 3 + sec²θ + cosec²θ − 2 − sec²θ × cosec²θ

⇒ (1 + sec²θ + cosec²θ − sec²θ × cosec²θ)

Now let’s simplify by assuming θ = 45° (a valid identity angle where all values are defined)

⇒ sec 45° = √2, so sec²θ = 2

⇒ cosec 45° = √2, so cosec²θ = 2

⇒ Expression = 1 + 2 + 2 − (2 × 2)

⇒ 1 + 4 − 4 = 1

∴ The value of the expression is 1.

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