Question

How do you prove the identity $\dfrac{1-\sin x}{\cos x}=\dfrac{\cos x}{1+\sin x}$ ?

Answer 1

Hint: Consider either the LHS or the RHS of the given question. Start by multiplying and dividing the expression by either $(1-\sin x)$ or $(1+\sin x)$ for LHS and RHS respectively. Simplify the terms using the trigonometric identity ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ and cancelling common terms.

Complete Step by Step Solution:
The identity given in the question is $\dfrac{1-\sin x}{\cos x}=\dfrac{\cos x}{1+\sin x}$ .
We can prove the identity by proving LHS = RHS
RHS: $\dfrac{\cos x}{1+\sin x}$
Multiplying and dividing the above expression by $(1-\sin x)$
$=\dfrac{\cos x}{1+\sin x}\times \dfrac{1-\sin x}{1-\sin x}$
Simplifying
$=\dfrac{\cos x(1-\sin x)}{(1+\sin x)(1-\sin x)}$
Using the formula $(a-b)(a+b)={{a}^{2}}-{{b}^{2}}$ , the denominator can be written as
$=\dfrac{\cos x(1-\sin x)}{1-{{\sin }^{2}}x}$
We know that ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ . Rearranging the terms of this identity, $1-{{\sin }^{2}}x={{\cos }^{2}}x$ . Substituting the value in the denominator,
$=\dfrac{\cos x(1-\sin x)}{{{\cos }^{2}}x}$
Cancelling the common terms in the numerator and the denominator, we get
$\Rightarrow \dfrac{1-\sin x}{\cos x}$ = LHS
Therefore, LHS=RHS. Hence, proved.

Note:
Alternate Method:
The identity given in the question is $\dfrac{1-\sin x}{\cos x}=\dfrac{\cos x}{1+\sin x}$ .
We can prove the identity by proving LHS = RHS
LHS: $\dfrac{1-\sin x}{\cos x}$
Multiplying and dividing the above expression by $(1+\sin x)$
$=\dfrac{1-\sin x}{\cos x}\times \dfrac{1+\sin x}{1+\sin x}$
Simplifying
$=\dfrac{1-\sin x(1+\sin x)}{\cos x(1+\sin x)}$
Using the formula $(a-b)(a+b)={{a}^{2}}-{{b}^{2}}$ , the numerator can be written as
$=\dfrac{1-{{\sin }^{2}}x}{\cos x(1+\sin x)}$
We know that ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ . Rearranging the terms of this identity, $1-{{\sin }^{2}}x={{\cos }^{2}}x$ . Substituting the value in the numerator,
$=\dfrac{{{\cos }^{2}}x}{\cos x(1+\sin x)}$
Cancelling the common terms in the numerator and the denominator, we get
$\Rightarrow \dfrac{\cos x}{1+\sin x}=RHS$
Therefore, LHS=RHS. Hence, proved.