Question

The value of (secA + tanA) (1−sinA) is equal to:
A) \[{{\tan }^{2}}A\]
B) \[{{\sin }^{2}}A\]
C) \[\cos A\]
D) $\sin A$

Answer 1

Hint: For solving this question, first we convert the whole expression in terms of sinA and cosA by using the trigonometric relations $\sec A\text{ as }\dfrac{1}{\cos A}$ and $\tan A\text{ as }\dfrac{\sin A}{\cos A}$. Now, we solve the numerator by applying another trigonometric relation of ${{\sin }^{2}}A+{{\cos }^{2}}A=1$. After this, we can easily calculate the value of the expression.
Complete step-by-step answer:
According to the problem statement, we are given an expression as $\left( \sec A+\tan A \right)\left( 1-\sin A \right)$. Now, by using the reciprocal identities $\sec \theta =\dfrac{1}{\cos \theta }$ and $\tan \theta =\dfrac{\sin \theta }{\cos \theta }$ respectively, we get the simplified expression as:
$\Rightarrow \left( \dfrac{1}{\cos A}+\dfrac{\sin A}{\cos A} \right)\left( 1-\sin A \right)$
Since the denominator is same, so solving for the numerator, we get
$\Rightarrow \left( \dfrac{1+\sin A}{\cos A} \right)\left( 1-\sin A \right)$
Using the algebraic identity ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$, we can also write $\left( 1-\sin A \right)\left( 1-\sin A \right)$ as $1-{{\sin }^{2}}A$. So, further simplification yields
$\begin{align}
  & \Rightarrow \dfrac{\left( 1-\sin A \right)\left( 1+\sin A \right)}{\cos A} \\
 & \Rightarrow \dfrac{1-{{\sin }^{2}}A}{\cos A} \\
\end{align}$
As we know that the ${{\sin }^{2}}A+{{\cos }^{2}}A=1$. So, we can rewrite $1-{{\sin }^{2}}A$ as ${{\cos }^{2}}A$. Putting it in above expression, we get
$\begin{align}
  & \Rightarrow \dfrac{{{\cos }^{2}}A}{\cos A} \\
 & \Rightarrow \cos A \\
\end{align}$
Hence, the value of (secA + tanA) (1 – sinA) is equal to cosA.
Therefore, option (C) is correct.
Note: This problem could be alternatively solved by using the hit and trial method. We can assume some suitable value of A and match the given expression and option by putting A. Consider A as 0, we obtain the expression to be 1. Putting A as 0 in all options, only option C yields the correct answer. So, option (C) is correct and obtained without calculation.