Question

Prove that :
$(\sin \theta +\sec \theta {)}^2+(\cos \theta +\cos ec\theta {)}^2=(1+\sec \theta \cos ec\theta {)}^2$.

Answer 1

We have,
LHS = $(sin\theta +sec\theta {)}^2+(cos\theta +cosec\theta {)}^2$

$\Rightarrow LHS={(sin\theta +\frac{1}{cos\theta })}^2+{(cos\theta +\frac{1}{sin\theta })}^2$

$\Rightarrow LHS=si{n}^2\theta +\frac{1}{co{s}^2\theta }+\frac{2sin\theta }{cos\theta }+co{s}^2\theta +\frac{1}{si{n}^2\theta }+\frac{2cos\theta }{sin\theta }$

$\Rightarrow LHS=(si{n}^2\theta +co{s}^2\theta )+(\frac{1}{co{s}^2\theta }+\frac{1}{si{n}^2\theta })+\frac{2(si{n}^2\theta +co{s}^2\theta )}{sin\theta cos\theta }$

$\Rightarrow LHS=1+\frac{1}{si{n}^2\theta co{s}^2\theta }+\frac{2}{sin\theta cos\theta }$

$\Rightarrow LHS={(1+\frac{1}{sin\theta cos\theta })}^2=(1+sec\theta cosec\theta {)}^2=RHS.$