Question
Download Solution PDFIf tan2 θ = 1 - a2, then the value of sec θ + tan3 θ cosec θ is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
tan2 θ = 1 - a2
Concept Used:
Secθ = 1/cosθ, tanθ = sinθ/cosθ, sinθ = 1/cosecθ
Sec2θ – tan2θ = 1
Calculation:
Secθ = 1/cosθ, tanθ = sinθ/cosθ, sinθ = 1/cosecθ
Sec2θ – tan2θ = 1
We have to calculate, sec θ + tan3 θ cosec θ
\(\begin{array}{l} = \frac{1}{{\cos \theta }} + \frac{{si{n^3}\theta }}{{co{s^3}\theta }} \times \frac{1}{{sin\theta }}\\ = \frac{1}{{\cos \theta }}\left[ {1 + \frac{{si{n^2}\theta }}{{co{s^2}\theta }}} \right]\\ = \frac{1}{{\cos \theta }} \times \frac{1}{{co{s^2}\theta }} = \frac{1}{{co{s^3}\theta }} \end{array}\)
Now, given that,
tan2θ = 1 – a2
⇒ sec2θ – 1 = 1 – a2
\(\begin{array}{l} \Rightarrow \frac{1}{{co{s^2}\theta }} = 2 - {a^2}\\ \Rightarrow \frac{1}{{co{s^3}\theta }} = {\left[ {2 - {a^2}} \right]^{\frac{3}{2}}} \end{array}\)
∴ Option 3 is the correct answer.
Last updated on Jun 11, 2025
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