Question

The value of $\tan \left( 225{}^\circ \right)\cot \left( 405{}^\circ \right)+\tan \left( 765{}^\circ \right)$ is
[a] 1
[b] 2
[c] 3
[d] 0

Answer 1

- Hint: Use the fact that$\forall n\in \mathbb{Z},\tan \left( n\pi +x \right)=\tan x,\cot \left( n\pi +x \right)=\cot x,\tan \left( \dfrac{\pi }{2}+x \right)=-\cot x,\tan \left( \dfrac{\pi }{2}-x \right)=\cot x,\cot \left( \dfrac{\pi }{2}+x \right)=-\tan x$ and $\cot \left( \dfrac{\pi }{2}-x \right)=\tan x$. Use $\tan x\cot x=1$ and $\tan \left( \dfrac{\pi }{4} \right)=1$.

Complete step-by-step answer:

Trigonometric ratios:
There are six trigonometric ratios defined on an angle of a right-angled triangle, viz sine, cosine, tangent, cotangent, secant and cosecant.
The sine of an angle is defined as the ratio of the opposite side to the hypotenuse. It is written as sinx
The cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse. It is written as cos x.
The tangent of an angle is defined as the ratio of the opposite side to the adjacent side. It is written as tanx.
The cotangent of an angle is defined as the ratio of the adjacent side to the opposite side.It is written as cotx.
The secant of an angle is defined as the ratio of the hypotenuse to the adjacent side. It is written as secx.
The cosecant of an angle is defined as the ratio of the hypotenuse to the adjacent side. It is written as cosec x.
Observe that sine and cosecant are multiplicative inverses of each other, cosine and secant are multiplicative inverses of each other, and tangent and cotangent are multiplicative inverses of each other.
We know that $x{}^\circ =\dfrac{\pi x}{180}$ radians.
Hence we have
$\begin{aligned} & 225{}^\circ =\dfrac{225}{180}\pi =\dfrac{5\pi }{4} \\\ & 405{}^\circ =\dfrac{405}{180}\pi =\dfrac{9\pi }{4} \\\ & 765=\dfrac{17\pi }{4} \\\ \end{aligned}$
Hence the given expression becomes,
$\begin{aligned} & \tan \left( \dfrac{5\pi }{4} \right)\cot \left( \dfrac{9\pi }{4} \right)+\tan \left( \dfrac{17\pi }{4} \right) \\\ & =\tan \left( \pi +\dfrac{\pi }{4} \right)\cot \left( 2\pi +\dfrac{\pi }{4} \right)+\tan \left( 4\pi +\dfrac{\pi }{4} \right) \\\ \end{aligned}$
We know that $\tan \left( n\pi +x \right)=\tan x,\cot \left( n\pi +x \right)=\cot x,n\in \mathbb{Z}$
Using the above formula, we get

$\tan \left( 225{}^\circ \right)\cot \left( 405{}^\circ \right)+\tan \left( 765{}^\circ \right)=\tan \left( \dfrac{\pi }{4} \right)\cot \left( \dfrac{\pi }{4} \right)+\tan \left( \dfrac{\pi }{4} \right)$
We know that $\tan x\cot x=1$ and $\tan \left( \dfrac{\pi }{4} \right)=1$
Using the above formula, we get
$\tan \left( 225{}^\circ \right)\cot \left( 405{}^\circ \right)+\tan \left( 765{}^\circ \right)=1+1=2$
Hence option [b] is correct.

Note: Rule for converting $T\left( n\dfrac{\pi }{2}\pm x \right),n\in \mathbb{Z}$ to $T\left( x \right)$, where T is any trigonometric ratio.
If n is even the final expression will be of T. If n is odd, the final expression will contain the complement of T.
The complement of sin is cos and vice versa
The complement of tan is cot and vice versa
The complement of cosec is sec and vice versa.
Sign of the final expression is determined by the quadrant in which $n\dfrac{\pi }{2}\pm x$ falls.
Keeping the above points in consideration, we have
$\tan \left( \pi +x \right)=\tan \left( 2\dfrac{\pi }{2}+x \right)$
Now 2 is even, hence the final expression will be of tan x.
Also, $\pi +x$ falls in the third quadrant in which tanx is positive
Hence, we have
$\tan \left( \pi +x \right)=\tan x$
[2] The sign of various trigonometric ratios in various quadrants can be remembered by the mnemonic
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A stands for All are positive in the first quadrant
S stands for sine (and its reciprocal) are positive in the second quadrant
T stands for tangent(and its reciprocal) are positive in the third quadrant
C stands for cosine(and its reciprocal) are positive in the fourth quadrant