JEE Main & Advanced Mathematics Trigonometric Identities Maximum and Minimum Value of a \[\mathbf{cos}\,\,\mathbf{\theta }\,\,\mathbf{+}\,\mathbf{b}\,\,\mathbf{sin}\,\,\mathbf{\theta }\]

Let  \[a=r\cos \alpha \]    .....(i)   and   \[b=r\sin \alpha \]         .....(ii)

Squaring and adding (i) and (ii), then \[{{a}^{2}}+{{b}^{2}}={{r}^{2}}\] or, \[r=\sqrt{{{a}^{2}}+{{b}^{2}}}\]

\[\therefore \]          \[a\sin \theta +b\cos \theta =r(\sin \theta \cos \alpha +\cos \theta \sin \alpha )=r\sin (\theta +\alpha )\]

But \[-1\le \sin \theta <1\] So, \[-1\le \sin (\theta +\alpha )\le 1\];

Then \[-r\le r\sin (\theta +\alpha )\le r\]

Hence, \[\sqrt{2}-1\]

Then the greatest and least values of \[a\sin \theta +b\cos \theta \] are respectively \[\sqrt{{{a}^{2}}+{{b}^{2}}}\] and \[-\sqrt{{{a}^{2}}+{{b}^{2}}}\].

Therefore, \[{{\sin }^{2}}x+c\text{ose}{{\text{c}}^{\text{2}}}x\ge 2,\] for every real \[x\].

\[{{\cos }^{2}}x+{{\sec }^{2}}x\ge 2,\] for every real \[x\].

\[{{\tan }^{2}}x+{{\cot }^{2}}x\ge 2\], for every real \[x\].