Question

Show that $(\sin \theta + \cos \theta )(\tan \theta + \cot \theta ) = \sec \theta + \cos ec\theta $

Answer 1

Hint: According to given in the question we have to show that $(\sin \theta + \cos \theta)(\tan \theta + \cot \theta ) = \sec \theta + \cos ec\theta $ so, we will solve the left hand side term
of the given expression which is $(\sin \theta + \cos \theta )(\tan \theta + \cot \theta )$
To solve L.H.S. first of all we will try to make the term $\tan \theta $ in form of $\sin \theta $ and $\cos \theta $ with the help of formula as given below:

Formula used: $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}.................(1)$
Now, same as we will try to make the term $\cot \theta $ in form of $\sin \theta $ and $\cos \theta$ with the help of formula as given below:
$\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}.................(2)$
After obtaining the expression in form of $\sin \theta $ and $\cos \theta $ we will take the L.C.M. to solve it and after L.C.M. we will obtain the terms $\sin \theta $ and $\cos \theta $ in their square form which can be solved with the help of the formula as given below:
${\sin ^2}\theta + {\cos ^2}\theta = 1...............(3)$

Complete step-by-step answer:
Step 1: First of all we have to make the term $\tan \theta $ as given in the L.H.S. of the expression in the form of $\sin \theta $ and $\cos \theta $ with the help of formula (1) as mentioned in the solution
hint.
$ = (\sin \theta + \cos \theta )\left( {\dfrac{{\sin \theta }}{{\cos \theta }} + \cot \theta } \right)$
Step 2: Same as, to make the term $\cot \theta $ as given in the L.H.S. of the expression in form of $\sin \theta $ and $\cos \theta $with the help of formula (2) as mentioned in the solution hint.
$ = (\sin \theta + \cos \theta )\left( {\dfrac{{\sin \theta }}{{\cos \theta }} + \dfrac{{\cos \theta }}{{\sin\theta }}} \right)$
Step 3: Now, we have to solve the trigonometric expression as obtained in step 2.
$ = (\sin \theta + \cos \theta )\left( {\dfrac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{\sin \theta \cos\theta }}} \right)$
Step 4: Now, to solve the expression as obtained in step 3 we have to use the formula (3) as mentioned in the solution hint.
$ = (\sin \theta + \cos \theta )\left( {\dfrac{1}{{\sin \theta \cos \theta }}} \right)$
Step 5: Now, we have to multiply and divide the terms of the expression as obtained in the step 4.
$
= \dfrac{{\sin \theta + \cos \theta }}{{\sin \theta \cos \theta }} \\
= \dfrac{{\sin \theta }}{{\sin \theta \cos \theta }} + \dfrac{{\cos \theta }}{{\sin \theta \cos \theta }} \\
$
On solving the obtained expression,
$ = \dfrac{1}{{\cos \theta }} + \dfrac{1}{{\sin \theta }}$…………………..(4)
Step 6: As we know that, $\sec \theta = \dfrac{1}{{\cos \theta }}$ and $\cos ec\theta =
\dfrac{1}{{\sin \theta }}$ hence, substituting in the expression (4) as obtained in the step 5.
$ = \sec \theta + \cos ec\theta $

Hence, with the help of the formula (1), (2), and (3) we have proved that $(\sin \theta + \cos \theta)(\tan \theta + \cot \theta ) = \sec \theta + \cos ec\theta $

Note: We can also solve the given expression by multiplying each term of the given expression but it will lead us to lots of difficult calculations.
To make the calculations easy we have to use the formula ${\sin ^2}\theta + {\cos ^2}\theta
= 1$ to eliminate the terms like ${\sin ^2}\theta $ and ${\cos ^2}\theta $.