Question

Prove that ${\left( {\sin A + \operatorname{cosec} A} \right)^2} + {\left( {\cos A + \sec A} \right)^2} = 7 + {\tan ^2}A + {\cot ^2}A$ .

Answer 1

Hint:
Firstly, take the L.H.S. of the given equation into consideration.
Then, expand the terms ${\left( {\sin A + \operatorname{cosec} A} \right)^2}$ and ${\left( {\cos A + \sec A} \right)^2}$ .
Finally, apply the properties of trigonometry as required in the L.H.S. of the question to prove it equal to the R.H.S. of the question.

Complete step by step solution:
To prove ${\left( {\sin A + \operatorname{cosec} A} \right)^2} + {\left( {\cos A + \sec A} \right)^2} = 7 + {\tan ^2}A + {\cot ^2}A$ , we will take L.H.S. and solve it further to prove it equal to R.H.S.
 $\therefore $ L.H.S. $ = {\left( {\sin A + \operatorname{cosec} A} \right)^2} + {\left( {\cos A + \sec A} \right)^2}$
Now, expanding ${\left( {\sin A + \operatorname{cosec} A} \right)^2}$ and ${\left( {\cos A + \sec A} \right)^2}$ .
 $\therefore $ L.H.S. $ = {\sin ^2}A + {\operatorname{cosec} ^2}A + 2\sin A\operatorname{cosec} A + {\cos ^2}A + {\sec ^2}A + 2\cos A\sec A$
Also, ${\sin ^2}A + {\cos ^2}A = 1$ , $\sin A\operatorname{cosec} A = 1$ and $\cos A\sec A = 1$ .
 $\therefore $ L.H.S. $ = 1 + 2 + 2 + {\operatorname{cosec} ^2}A + {\sec ^2}A$
And ${\operatorname{cosec} ^2}A = 1 + {\cot ^2}A$ and ${\sec ^2}A = 1 + {\tan ^2}A$ .
 $\therefore $ L.H.S. $ = 5 + 1 + {\cot ^2}A + 1 + {\tan ^2}A$
   $ = 7 + {\tan ^2}A + {\cot ^2}A$
  $ = $ R.H.S.

Hence, proved that ${\left( {\sin A + \operatorname{cosec} A} \right)^2} + {\left( {\cos A + \sec A} \right)^2} = 7 + {\tan ^2}A + {\cot ^2}A$.

Note:
Some properties of trigonometry:
 \[
  {\sin ^2}A + {\cos ^2}A = 1 \\
  {\tan ^2}A + {\sec ^2}A = 1 \\
  \cos e{c^2}A + {\cot ^2} = 1 \\
  \sin A\cos ecA = 1 \\
  \cos A\sec A = 1 \\
  \tan A\cot A = 1 \\
 \]