What is the least value of \tan^2\theta + \cot^2 \theta+ \sin^2 \theta + \cos^2 \theta + \sec^2 \theta+ \cosec ^2\theta ?

SSC CGL 9 Aug 2017 Shift 1 Paper

Question : 71 of 100

Marks: +1, -0

What is the least value of

\tan^2θ + \cot^2 θ+ \sin^2 θ + \cos^2 θ

+ \sec^2 θ+ \cosec ^2θ\ ?

Solution:

(\tan^2θ + \cot^2 θ)+ (\sin^2 θ + \cos^2 θ)

+ (\sec^2 θ+ \cosec ^2θ)

=2 +1 +(\sec^2 θ+ \cosec ^2θ)

(∵\sin^2 θ + \cos^2 θ=1)

∴\tan^2θ + \cot^2 θ=2

=3+(1/{\cos^2θ}+1/{\sin^2θ})

=3+({\sin^2 θ + \cos^2 θ}/{\sin^2 θ. \cos^2 θ})

∵ Minimum value of

(\sinθ.\cosθ)^n=(1/2)^n

∴

(\sinθ.\cosθ)^2=(1/2)^2=1/4

=3+(1/{1/4})=3+4=7

Go to Question:

Prev Question

Next Question