# Answer the following: If sec θ = 2 and 3π2<θ<2π then evaluate 1+tanθ+cosecθ1+cotθ-cosecθ - Mathematics and Statistics

Sum

If sec θ = sqrt(2) and (3pi)/2 < theta < 2pi then evaluate (1 + tantheta + "cosec"theta)/(1 + cottheta - "cosec"theta)

#### Solution

Given, sec θ = sqrt(2)

We know that,

tan2θ = sec2θ – 1

= (sqrt(2))^2 - 1

= 2 – 1 = 1

∴ tan θ = ± 1

Since (3pi)/2 < theta < 2pi,

θ lies in the 4th quadrant.

∴ tan θ < 0

∴ tan θ = – 1

cot θ = 1/(tan theta) = – 1

cos θ = 1/sec theta = 1/sqrt(2)

tan θ = sintheta/costheta

∴ sin θ = tan θ cos θ

= (-1)(1/sqrt(2))

= -1/sqrt(2)

∴ cosec θ = 1/sin theta = -sqrt(2)

∴ (1 + tan theta + "cosec" theta)/(1 + cot theta - "cosec" theta)

= (1 - 1 - sqrt(2))/(1 - 1 + sqrt(2))

= (-sqrt(2))/sqrt(2)

= – 1

Concept: Signs of Trigonometric Functions in Different Quadrants
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#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 2 Trigonometry - 1
Miscellaneous Exercise 2 | Q 9 | Page 33