Question

Prove that, $\sin x + \cos x\cot x = \csc x$

Answer 1

All trigonometric functions are defined for a right-angled triangle. To verify the trigonometric identities or equations, either we use the basic formulas that define the individual functions or we use the defined identities to solve the same. For a right-angled triangle with hypotenuse as “h”, perpendicular as “p” and base as “b”, we define $\sin \theta = \dfrac{p}{h},\cos \theta = \dfrac{b}{h},\tan \theta = \dfrac{p}{b}$

Complete step by step answer:
Amongst all the ways to prove this question, the easiest is to prove that the left-hand side is equal to the right-hand side. ($LHS = RHS$)
For any right-angled triangle, $\sin \theta = \dfrac{p}{h},\cos \theta = \dfrac{b}{h},\tan \theta = \dfrac{p}{b},\csc \theta = \dfrac{h}{p},\sec \theta = \dfrac{h}{b},\cot \theta = \dfrac{b}{p}$
Where hypotenuse is denoted by “h”, perpendicular is denoted by “p” and base is denoted by “b”. We know that,
$LHS = \sin x + \cos x\cot x$
$\dfrac{p}{h} + \left( {\dfrac{b}{h} \times \dfrac{b}{p}} \right)$………….(putting the values of the functions)
$\dfrac{{{p^2} + {b^2}}}{{hp}} = \dfrac{{{h^2}}}{{hp}} = \dfrac{h}{p}$……..(Since, sum of square of perpendicular and base of a right-angled triangle is equal to the square of the hypotenuse.)
$\Rightarrow LHS = \dfrac{h}{p}$
Now, $\csc \theta = \dfrac{h}{p}$
Hence, $LHS = \csc x = RHS$
Thus, the question is verified.

Alternate method:
We know that $\cot x = \dfrac{{\cos x}}{{\sin x}}$. Using this on the LHS,
$LHS = \sin x + \cos x \times \dfrac{{\cos x}}{{\sin x}} = \sin x + \dfrac{{{{\cos }^2}x}}{{\sin x}}$
Taking the LCM,
$LHS = \dfrac{{{{\sin }^2}x + {{\cos }^2}x}}{{\sin x}} = \dfrac{1}{{\sin x}}$……( since we know that, ${\sin ^2}x + {\cos ^2}x = 1$)
Now, $\csc x = \dfrac{1}{{\sin x}}$.
Therefore, LHS = RHS.

Note: Each of the six trigonometric functions is equal to its co-function evaluated at the complementary angle. Periodicity of trig functions: Sine, cosine, secant, and cosecant have period 2π while tangent and cotangent have period π. Sine, tangent, cotangent, and cosecant are odd functions while cosine and secant are even functions.