Question

If the value of the trigonometric expression $\cos A + {\cos ^2}A = 1$ then ${\sin ^2}A + {\sin ^4}A = 1$ is true or false?
(a) True
(b) False
(c) Ambiguous
(d) Data insufficient

Answer 1

Hint – In this question use trigonometric identity that ${\cos ^2}A = 1 - {\sin ^2}A$ and substitute it in $\cos A + {\cos ^2}A = 1$, then square both sides and again apply the trigonometric identity this will help getting the required entity.

Complete step-by-step answer:
Given equation
$\cos A + {\cos ^2}A = 1$
Now as we know that ${\cos ^2}A = 1 - {\sin ^2}A$ so substitute this value in above equation we have,
$\Rightarrow \cos A + 1 - {\sin ^2}A = 1$
$\Rightarrow \cos A = {\sin ^2}A$
Now squaring on both sides we have,
$\Rightarrow {\left( {\cos A} \right)^2} = {\left( {{{\sin }^2}A} \right)^2}$
Now simplify it we have,
$\Rightarrow \left( {{{\cos }^2}A} \right) = \left( {{{\sin }^4}A} \right)$
Now again using the property which is explained above we have,
$\Rightarrow \left( {1 - {{\sin }^2}A} \right) = \left( {{{\sin }^4}A} \right)$
$\Rightarrow {\sin ^2}A + {\sin ^4}A = 1$
Which is the required given second equation.
Hence the second given equation is true.
So this is the required answer.
Hence option (A) is the correct answer.

Note – It is always advised to remember basic trigonometric identities like ${\sin ^2}A + {\cos ^2}A = 1$ and $1 + {\tan ^2}A = {\sec ^2}A$, along with $\cos 2A = {\cos ^2}A - {\sin ^2}A$, as these identities helps saving a lot of time while solving the problems of this kind. Such types of problems start by manipulating the given expression by using identities, rather than directly manipulating the required expression.