\displaystyle \textbf{Question 1: }\text{The value of }(\sin30^\circ+\cos30^\circ)-(\sin60^\circ+\cos60^\circ)\text{ is }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle (\sin30^\circ+\cos30^\circ)-(\sin60^\circ+\cos60^\circ)
\displaystyle =\left(\frac12+\frac{\sqrt3}{2}\right)-\left(\frac{\sqrt3}{2}+\frac12\right)=0
\displaystyle \therefore \text{The required value is }0.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{The value of }\frac{\tan30^\circ}{\cot60^\circ}\text{ is }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle \frac{\tan30^\circ}{\cot60^\circ}=\frac{\frac{1}{\sqrt3}}{\frac{1}{\sqrt3}}=1
\displaystyle \therefore \text{The required value is }1.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{The value of }(\sin45^\circ+\cos45^\circ)^3\text{ is }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle (\sin45^\circ+\cos45^\circ)^3
\displaystyle =\left(\frac{1}{\sqrt2}+\frac{1}{\sqrt2}\right)^3=(\sqrt2)^3=2\sqrt2
\displaystyle \therefore \text{The required value is }2\sqrt2.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{The value of }(\sin30^\circ+\cos30^\circ)^2-(\sin60^\circ-\cos60^\circ)^2\text{ is }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle =\left(\frac12+\frac{\sqrt3}{2}\right)^2-\left(\frac{\sqrt3}{2}-\frac12\right)^2
\displaystyle =\left[\left(\frac12+\frac{\sqrt3}{2}\right)+\left(\frac{\sqrt3}{2}-\frac12\right)\right]
\displaystyle \qquad\times\left[\left(\frac12+\frac{\sqrt3}{2}\right)-\left(\frac{\sqrt3}{2}-\frac12\right)\right]
\displaystyle =\sqrt3\times1=\sqrt3
\displaystyle \therefore \text{The required value is }\sqrt3.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{The value of }(\cos^2 23^\circ-\sin^2 67^\circ)\text{ is }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle \sin67^\circ=\cos(90^\circ-67^\circ)=\cos23^\circ
\displaystyle \therefore \cos^2 23^\circ-\sin^2 67^\circ
\displaystyle =\cos^2 23^\circ-\cos^2 23^\circ=0
\displaystyle \therefore \text{The required value is }0.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{The value of the expression }\left\{\frac{\sin^2 22^\circ+\sin^2 68^\circ}{\cos^2 22^\circ+\cos^2 68^\circ}+\sin^2 63^\circ+\cos63^\circ\sin27^\circ\right\}\text{ is }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle \sin68^\circ=\cos22^\circ,\quad\cos68^\circ=\sin22^\circ,\quad\sin27^\circ=\cos63^\circ
\displaystyle \therefore \frac{\sin^2 22^\circ+\sin^2 68^\circ}{\cos^2 22^\circ+\cos^2 68^\circ}
\displaystyle =\frac{\sin^2 22^\circ+\cos^2 22^\circ}{\cos^2 22^\circ+\sin^2 22^\circ}=1
\displaystyle \sin^2 63^\circ+\cos63^\circ\sin27^\circ
\displaystyle =\sin^2 63^\circ+\cos^2 63^\circ=1
\displaystyle \therefore \text{The required value is }1+1=2.
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Given that }\sin\alpha=\frac12\text{ and }\cos\beta=\frac12,\text{ then }\alpha+\beta=\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle \sin\alpha=\frac12=\sin30^\circ\quad\therefore\quad\alpha=30^\circ
\displaystyle \cos\beta=\frac12=\cos60^\circ\quad\therefore\quad\beta=60^\circ
\displaystyle \therefore \alpha+\beta=30^\circ+60^\circ=90^\circ.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{Given that }\sin(\alpha-\beta)=\frac12\text{ and }\cos(\alpha+\beta)=\frac12,\text{ then}
\displaystyle \alpha=\underline{\hspace{1.5cm}},\quad\beta=\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle \sin(\alpha-\beta)=\frac12=\sin30^\circ
\displaystyle \therefore \alpha-\beta=30^\circ
\displaystyle \cos(\alpha+\beta)=\frac12=\cos60^\circ
\displaystyle \therefore \alpha+\beta=60^\circ
\displaystyle \text{Adding the two equations, }2\alpha=90^\circ
\displaystyle \therefore \alpha=45^\circ
\displaystyle \text{Substituting in }\alpha+\beta=60^\circ,\quad\beta=15^\circ
\displaystyle \therefore \alpha=45^\circ,\qquad\beta=15^\circ.
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{If }4\tan\theta=3,\text{ then }\frac{4\sin\theta-\cos\theta}{4\sin\theta+\cos\theta}\text{ is equal to }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle 4\tan\theta=3\quad\therefore\quad\tan\theta=\frac34
\displaystyle \frac{4\sin\theta-\cos\theta}{4\sin\theta+\cos\theta}
\displaystyle =\frac{4\tan\theta-1}{4\tan\theta+1}
\displaystyle =\frac{4\left(\frac34\right)-1}{4\left(\frac34\right)+1}
\displaystyle =\frac{3-1}{3+1}=\frac24=\frac12
\displaystyle \therefore \text{The required value is }\frac12.
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{If }\cos5\theta=\sin\theta\text{ and }5\theta<90^\circ,\text{ then the value of }\tan3\theta\text{ is }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle \cos5\theta=\sin\theta=\cos(90^\circ-\theta)
\displaystyle \therefore 5\theta=90^\circ-\theta
\displaystyle 6\theta=90^\circ
\displaystyle \therefore \theta=15^\circ
\displaystyle \therefore \tan3\theta=\tan45^\circ=1
\displaystyle \therefore \text{The required value is }1.
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{The value of }\sec^2 60^\circ-\tan^2 60^\circ\text{ is }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle \sec60^\circ=2,\qquad\tan60^\circ=\sqrt3
\displaystyle \sec^2 60^\circ-\tan^2 60^\circ
\displaystyle =(2)^2-(\sqrt3)^2=4-3=1
\displaystyle \therefore \text{The required value is }1.
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{If }A+B=90^\circ,\text{ then the value of }\tan^2 A-\cot^2 B\text{ is }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle A+B=90^\circ
\displaystyle \therefore A=90^\circ-B
\displaystyle \tan A=\tan(90^\circ-B)=\cot B
\displaystyle \therefore \tan^2 A-\cot^2 B=\cot^2 B-\cot^2 B=0
\displaystyle \therefore \text{The required value is }0.
\displaystyle \\

\displaystyle \textbf{Question 13: }\text{The value of }\cos1^\circ\cos2^\circ\cos3^\circ\ldots\cos120^\circ\text{ is }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle \text{The given product contains the factor }\cos90^\circ.
\displaystyle \cos90^\circ=0
\displaystyle \therefore \cos1^\circ\cos2^\circ\cos3^\circ\ldots\cos120^\circ=0.
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{If }\tan\theta+\cot\theta=2\text{ and }0^\circ<\theta<90^\circ,\text{ then}
\displaystyle \tan^{10}\theta+\cot^{10}\theta\text{ is equal to }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle \text{Let }\tan\theta=x,\quad\therefore\quad\cot\theta=\frac1x.
\displaystyle x+\frac1x=2
\displaystyle x^2+1=2x
\displaystyle (x-1)^2=0
\displaystyle \therefore x=1
\displaystyle \therefore \tan\theta=1,\qquad\cot\theta=1
\displaystyle \tan^{10}\theta+\cot^{10}\theta=1^{10}+1^{10}=2
\displaystyle \therefore \text{The required value is }2.
\displaystyle \\

\displaystyle \textbf{Question 15: }\text{If }\sin\theta+\cos\theta=1\text{ and }0^\circ\leq\theta\leq90^\circ,\text{ then the possible values}
\displaystyle \text{of }\theta\text{ are }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle \sin\theta+\cos\theta=1
\displaystyle \text{Squaring both sides,}
\displaystyle \sin^2\theta+\cos^2\theta+2\sin\theta\cos\theta=1
\displaystyle 1+2\sin\theta\cos\theta=1
\displaystyle 2\sin\theta\cos\theta=0
\displaystyle \therefore \sin\theta=0\text{ or }\cos\theta=0
\displaystyle \therefore \theta=0^\circ\text{ or }90^\circ.
\displaystyle \\

\displaystyle \textbf{Question 16: }\text{If }\sin(A+B)=\cos(A-B)=\frac{\sqrt3}{2},\text{ then }\cot2A=\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle \sin(A+B)=\frac{\sqrt3}{2}=\sin60^\circ
\displaystyle \therefore A+B=60^\circ
\displaystyle \cos(A-B)=\frac{\sqrt3}{2}=\cos30^\circ
\displaystyle \therefore A-B=30^\circ
\displaystyle \text{Adding the two equations, }2A=90^\circ
\displaystyle \therefore \cot2A=\cot90^\circ=0.
\displaystyle \\

\displaystyle \textbf{Question 17: }\text{If }\triangle ABC\text{ is an isosceles right triangle right angled at }B,\text{ then}
\displaystyle \frac{\tan A+\cot C}{\cot A+\cot C}=\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle \angle B=90^\circ\text{ and }\angle A=\angle C
\displaystyle \therefore A=C=45^\circ
\displaystyle \frac{\tan A+\cot C}{\cot A+\cot C}
\displaystyle =\frac{\tan45^\circ+\cot45^\circ}{\cot45^\circ+\cot45^\circ}
\displaystyle =\frac{1+1}{1+1}=1
\displaystyle \therefore \text{The required value is }1.
\displaystyle \\

\displaystyle \textbf{Question 18: }\text{If }\alpha+\beta=90^\circ\text{ and }\alpha=\frac{\beta}{2},\text{ then }\tan\alpha\tan\beta=\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle \alpha+\beta=90^\circ,\qquad\alpha=\frac{\beta}{2}
\displaystyle \frac{\beta}{2}+\beta=90^\circ
\displaystyle \frac{3\beta}{2}=90^\circ
\displaystyle \therefore \beta=60^\circ,\qquad\alpha=30^\circ
\displaystyle \tan\alpha\tan\beta=\tan30^\circ\tan60^\circ
\displaystyle =\frac{1}{\sqrt3}\times\sqrt3=1
\displaystyle \therefore \text{The required value is }1.
\displaystyle \\

\displaystyle \textbf{Question 19: }\text{If in a triangle }ABC,\text{ angles }A\text{ and }B\text{ are complementary, then the value of }\cot C
\displaystyle \text{is }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle A+B=90^\circ
\displaystyle A+B+C=180^\circ
\displaystyle 90^\circ+C=180^\circ
\displaystyle \therefore C=90^\circ
\displaystyle \cot C=\cot90^\circ=0
\displaystyle \therefore \text{The required value is }0.
\displaystyle \\

\displaystyle \textbf{Question 20: }\text{The value of }\tan25^\circ\tan10^\circ\tan80^\circ\tan65^\circ\text{ is }\underline{\hspace{1.5cm}}.
\displaystyle \text{Answer:}
\displaystyle \tan65^\circ=\cot25^\circ,\qquad\tan80^\circ=\cot10^\circ
\displaystyle \therefore \tan25^\circ\tan10^\circ\tan80^\circ\tan65^\circ
\displaystyle =(\tan25^\circ\cot25^\circ)(\tan10^\circ\cot10^\circ)
\displaystyle =1\times1=1
\displaystyle \therefore \text{The required value is }1.
\displaystyle \\

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