Question
Download Solution PDFThe value of tan2θ + cot2θ - sec2θ cosec2θ is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
tan2θ + cot2θ - sec2θ cosec2θ
Concept used:
1. tanα = sinα/cosα
2. cotα = 1/tanα
3. secα = 1/cosα
4. cosecα = 1/sinα
5. (a + b)2 - 2ab = a2 + b2
6. sin2α + cos2α = 1
Calculation:
tan2θ + cot2θ - sec2θ cosec2θ
⇒ \(\frac {sin^2θ}{cos^2θ} + \frac {cos^2θ}{sin^2θ} - \frac {1}{sin^2θ \times cos^2θ}\)
⇒ \(\frac {sin^4θ + cos^4θ - 1}{sin^2θ \times cos^2θ}\)
⇒ \(\frac {(sin^2θ + cos^2θ)^2 - 2sin^2θ cos^2θ - 1}{sin^2θ \times cos^2θ}\)
⇒ \(\frac {(1)^2 - 2sin^2θ cos^2θ - 1}{sin^2θ \times cos^2θ}\)
⇒ \(\frac {-2sin^2θ cos^2θ}{sin^2θ \times cos^2θ}\)
⇒ -2
∴ The required answer is -2.
Shortcut Trick
Use value putting method to solve this question,
Use θ = 45°
tan2θ + cot2θ - sec2θ cosec2θ
⇒ 12 + 12 - (√2)2(√2)2
⇒ 1 + 1 - 4
⇒ 2 - 4 = - 2
∴ The correct answer to this question is -2.
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