NumericalHard4 marks
Without using trigonometrical tables, evaluate: cosec² 57° - tan² 33° + cos 44° cosec 46° - √2 cos 45° - tan² 60°
Topic: Trigonometric identities
Previous-year question practice database
Answer
Exam answer
cosec²57° − tan²33° + cos44° × cosec46° − √2 cos45° − tan²60°
Since 57° and 33° are complementary: cosec57° = sec33°
Therefore: cosec²57° − tan²33° = sec²33° − tan²33° = 1
Also: sin46° = cos44°
Hence: cos44° × cosec46° = cos44° / sin46° = 1
Further: √2 cos45° = √2 × 1/√2 = 1
and: tan²60° = (√3)² = 3
Therefore: Required value = 1 + 1 − 1 − 3 = −2
Answer:
−2.
Explanation
Use complementary-angle identities together with sec²θ − tan²θ = 1, cos45° = 1/√2 and tan60° = √3.
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