Question 96

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

Solution

Expression : $$tan^2θ + cot^2θ + sin^2θ + cos^2θ + sec^2θ + cosec^2θ$$

Using, $$(sec^2\theta-tan^2\theta=1)$$ and $$(cosec^2\theta-cot^2\theta=1)$$

= $$(sin^2\theta+cos^2\theta)+tan^2\theta+cot^2\theta+(1+tan^2\theta)+(1+cot^2\theta)$$

= $$1+1+1+2tan^2\theta+2cot^2\theta$$

= $$3+2(tan^2\theta+cot^2\theta)$$

Minimum value of $$(tan^2\theta+cot^2\theta)=2$$

= $$3+2(2)=3+4=7$$

=> Ans - (D)

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