Question

What is the value of ${{\cos }^{2}}\theta $ in terms of $\sin \theta $?

Answer 1

Hint: We should know that this problem is from trigonometric ratios. Here we have been asked to deduce a relation and more specifically, write the value of ${{\cos }^{2}}\theta $ in terms of $\sin \theta $ or maybe as a function of $\sin \theta $. For that we have to first deduce a relation with only $\sin \theta $ and $\cos \theta $ in it. We can also use some trigonometric identities such as ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ to get the desired answer.

Complete step-by-step solution:
In the given question, we have been asked the relation of ${{\cos }^{2}}\theta $ in terms of $\sin \theta $. Now, following the rules of trigonometry, we have an identity which has both ${{\cos }^{2}}\theta $ and $\sin \theta $ in it, which is given as,
${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$
Rearranging the terms in this, we can say that,
${{\cos }^{2}}\theta =1-{{\sin }^{2}}\theta $
We know the identity given by $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$ . Here, if we check the RHS, we can see that it is in the form if ${{a}^{2}}-{{b}^{2}}$ , where a is 1 and b is $\sin \theta $.
Applying the identity, we get
${{\cos }^{2}}\theta =\left( 1+\sin \theta \right)\left( 1-\sin \theta \right)$
Hence, we have derived ${{\cos }^{2}}\theta $ in terms of $\sin \theta $.
We can also take another approach and check if it satisfies the given condition or not. We know that,
$\tan \theta =\dfrac{\sin \theta }{\cos \theta }$
Now, this can also be written as,
$\cos \theta =\dfrac{\sin \theta }{\tan \theta }$
Now when we square both sides, we can write it as
${{\cos }^{2}}\theta =\dfrac{{{\sin }^{2}}\theta }{{{\tan }^{2}}\theta }$
We can see that this is also another relation which has ${{\cos }^{2}}\theta $ and $\sin \theta $, but since it also contains $\tan \theta $, we will not consider it. We must follow the earlier approach to satisfy the condition given in the question.

Note: While solving this question, we should remember that trigonometry involves a lot of ratios and identities. But the main point that we have to consider is to choose an identity which includes $\sin \theta $ and $\cos \theta $ only and has no other trigonometric ratios.