Question

Choose the correct answer from the alternatives given.
If $0<\theta <{90}^{\circ }$ then what is the minimum value of $si{n}^2\theta +co{s}^2\theta +ta{n}^2\theta +co{t}^2\theta +se{c}^2\theta ++cose{c}^2\theta$ ?

• A. 10
• B. 5
• C. 6
• D. 7

Answer 1

The correct option is D 7

$sin^2 \theta + cos^2 \theta + tan^2 \theta + cot^2 \theta + sec^2 \theta + cosec^2 \theta$
= $1+ta{n}^2\theta +co{t}^2\theta +se{c}^2\theta +cose{c}^2\theta$
Minimum value of a $ta{n}^2\theta +bco{t}^2\theta =2\sqrt{ab}$
=1 + 2$\sqrt{1\times 1}+se{c}^2\theta +cose{c}^2\theta =1+2+se{c}^2\theta +cose{c}^2\theta$
Now, minimum value of a $se{c}^2\theta +bcose{c}^2\theta =(\sqrt{a}+\sqrt{b}{)}^2$
= 1 + 2 + $(\sqrt{1}+\sqrt{1}{)}^2=1+2+(2{)}^2$ = 1 + 2 + 4 = 7
Hence, the minimum value of expression is 7.