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Question
(1 + tan θ + sec θ) (1 + cot θ − cosec θ) = ______.
Options
0
1
2
-1
none of these
Solution
(1 + tan θ + sec θ) (1 + cot θ − cosec θ) = 2.
Explanation:
(1 + tan θ + sec θ) (1 + cot θ − cosec θ)
= `(1+ (sin theta)/(cos theta)+1/(costheta))(1+(costheta)/(sin theta)-1/(sin theta))`
= `((costheta+sintheta +1)/costheta)((sintheta+cos theta -1)/sintheta)`
= `((sintheta+costheta)^2-(1)^2)/(sinthetacostheta)`
= `(sin^2theta+cos^2 theta + 2sin theta cos theta -1)/(sinthetacostheta)`
= `(1+2sinthetacostheta -1)/(sinthetacostheta)`
= `(2sintheta costheta)/(sin theta costheta)`
= 2
Hence, alternative 2 is correct.
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