Evaluate $\cos^2{26^\circ}$ $+$ $\cos{64^\circ} \sin{26^\circ}$ $+$ $\dfrac{\tan{36^\circ}}{\cot{54^\circ}}$

In this trigonometric expression, the square of cosine of angle twenty six degrees is added to the product of the cosine of angle sixty four degrees and sine of angle twenty six degrees. The sum of them is added to the quotient of the tan of angle thirty six degrees by the cot of angle fifty four degrees.

$\cos^2{26^\circ}$ $+$ $\cos{64^\circ}.\sin{26^\circ}$ $+$ $\dfrac{\tan{36^\circ}}{\cot{54^\circ}}$

In this trigonometry problem, we have to find the given trigonometric expression by simplification.

Focus on simplifying the expression

In this trigonometric expression, three terms are added to express their sum. Let us try to add two terms firstly. So, focus on the first two terms in the given expression. In fact, the cosine squared of twenty six degrees is not known to us. Similarly, the cosine of angle sixty four degrees and sine of angle twenty six are also unknown to us. It seems, it is too tough to evaluate for us. However, it can be easily simplified by thinking logically.

$\cos^2{26^\circ}$ $+$ $\cos{64^\circ}.\sin{26^\circ}$ $+$ $\dfrac{\tan{36^\circ}}{\cot{54^\circ}}$

In second term, the sum of the angles of the trigonometric are complementary angles. Hence, sine function can be transformed into cosine function and vice-versa. In the first term, the cosine of angle twenty six is in square form. It indicates that the factor cosine of angle sixty four degrees in the second term should be converted into sine of complementary angle as per the cofunction identity of cosine. Then only it is possible to add the first two terms in the trigonometric expression.

$=\,\,\,$ $\cos^2{26^\circ}$ $+$ $\cos{(90^\circ-26^\circ)} \times \sin{26^\circ}$ $+$ $\dfrac{\tan{36^\circ}}{\cot{54^\circ}}$

$=\,\,\,$ $\cos^2{26^\circ}$ $+$ $\sin{26^\circ} \times \sin{26^\circ}$ $+$ $\dfrac{\tan{36^\circ}}{\cot{54^\circ}}$

$=\,\,\,$ $\cos^2{26^\circ}$ $+$ $\sin^2{26^\circ}$ $+$ $\dfrac{\tan{36^\circ}}{\cot{54^\circ}}$

Add the two terms in the expression

According to the Pythagorean identity of the sine and cosine functions, the sum of the squares of sine and cosine are equal to one.

$=\,\,\,$ $\cos^2{26^\circ}$ $+$ $\sin^2{26^\circ}$ $+$ $\dfrac{\tan{36^\circ}}{\cot{54^\circ}}$

$=\,\,\,$ $1+\dfrac{\tan{36^\circ}}{\cot{54^\circ}}$

Evaluate the trigonometric expression

Continue to think in the same logic to evaluate the simplified trigonometric expression.

$=\,\,\,$ $1+\dfrac{\tan{36^\circ}}{\cot{(90^\circ-36^\circ)}}$

The sum of the angles in the trigonometric functions are also complementary angles. Therefore, the tan of angles thirty six degrees can be converted into cot function. Similarly, the cot of angle fifty four degrees can also be converted into tan functions as per convenience. In this problem, the cotangent of fifty four degrees is converted into tan of complementary angle as per the cofunction identity of the cot function.

$=\,\,\,$ $1+\dfrac{\tan{36^\circ}}{\tan{36^\circ}}$

$=\,\,\,$ $1+\dfrac{\cancel{\tan{36^\circ}}}{\cancel{\tan{36^\circ}}}$

$=\,\,\,$ $1+1$

$=\,\,\,$ $2$