Question

cos^2a[1+tan^2a]+sin^2a[1+cot^2a]=2

Answer 1

Solution:-

$\rm\implies \cos^{2} a(1+\tan^{2} a) + \sin^2a(1+\cot^2a)=2$

Now multiply

$\rm\implies \cos^{2} a+ \cos^{2} a\tan^{2} a + \sin^2a+\sin^2a\cot^2a$

Using trigonometry identities

$\to\rm \tan a=\dfrac{\sin a}{\cos a}$

$\rm\to \cot a=\dfrac{\cos a}{\sin a}$

$\rm\to \sin x^{2} + \cos x^{2} = 1$

Now we get

$\rm\implies \cos^2 a + \cos^2 a\times \dfrac{\sin^2 a}{\cos ^2a}+\sin^2 a+\sin^2 a\times\dfrac{\cos^2 a}{\sin^2 a}$

$\rm\implies \cos^2 a +\cancel{ \cos^2 a}\times \dfrac{\sin^2 a}{\cancel{\cos ^2a}}+\sin^2 a+\cancel{\sin^2 a}\times\dfrac{\cos^2 a}{\cancel{\sin^2 a} }$

$\rm\implies \cos^2 a + {\sin^2 a}+\sin^2 a+{\cos^2 a}{}$

Using trigonometry identities

$\rm\implies1+1$

$\implies 2$

Hence proved

Definition of trigonometry

⇒ It is the branch of mathematics which deal with the measurement of angles and problems allied with angles