Consider the following statements : 1. 1+²θ/1+²θ=≤ft(1-θ/1-θ)² is true for all 0 2. θ=1/θ is true for θ=45^{} only Which of the above statements is/arecorrect ?

CDS 2015 PYQ with Answer — Consider the following statements : 1. 1+²θ/1+²θ=≤ft(1-θ/1-θ)² is tru…

2015 · Q36

Consider the following statements :

11+²θ/1+²θ=≤ft(1-θ/1-θ)² is true for all 0
2θ=1/θ is true for θ=45^{} only

Which of the above statements is/arecorrect ?

Elementary MathematicsSet Theory

A1 only
B2 only
CBoth 1 and 2
DNeither 1 nor 2

Explanation

Statement 1: LHS = (1 + tan^2)/(1 + cot^2) = tan^2, and RHS = ((1 - tan)/(1 - cot))^2 = (-tan)^2 = tan^2 for tan != 1, so it is true. Statement 2 is false since cot = 1/tan is an identity, not special to 45 deg. Hence 1 only, option (a).