Trigonometric Expressions and Identities: Problems with Solutions

Problems on trigonometric expressions: exact values, simplifying with the addition, double-angle and sum-to-product formulas, the value of an expression when tan α is known, conditional identities in a triangle and proofs of identities. Every problem has a step-by-step solution.

Problem 1

Prove that the value of the expression [tex]2\sin ^4\alpha+2\cos^4\alpha+\sin ^2 2\alpha[/tex] does not depend on [tex]\alpha[/tex], and find it.

Problem 2

Calculate [tex](\sin \alpha-\cos\alpha)(\sin \beta-\cos\beta)[/tex] if [tex]\sin (\alpha+\beta)=0. 8[/tex] and [tex]\cos(\alpha-\beta)=0. 3[/tex].

[tex]\frac{1}{2}[/tex]

[tex]\frac{11}{10}[/tex]

[tex]-\frac{1}{2}[/tex]

[tex]-\frac{11}{10}[/tex]

Problem 3

The angles [tex]A[/tex] and [tex]B[/tex] are such that [tex]\cos(A-B)=3\cos(A+B)[/tex] and [tex]\sin A\sin B\ne 0[/tex]. Find [tex]\cot A\times \cot B[/tex].

Problem 4

Find the value of the expression [tex]\frac{\cos 2\alpha}{1-\sin 2\alpha}[/tex] if [tex]\tan \alpha=\frac{3}{4}[/tex].

Problem 5

Without tables or a calculator, calculate [tex]\tan ^2 15^{\circ}+\cot ^2 15^{\circ}[/tex].

Problem 6

Simplify the expression [tex]\sin \alpha-2\sin \left(\frac{\alpha}{2}-15^{\circ}\right)\cos\left(\frac{\alpha}{2}+15^{\circ}\right)[/tex].

[tex]\frac{1}{2}[/tex]

[tex]-\frac{1}{2}[/tex]

[tex]\frac{\sqrt{3}}{2}[/tex]

[tex]2\sin \alpha-\frac{1}{2}[/tex]

Problem 7

Simplify the expression [tex]\frac{2\sin \alpha-\sin 2\alpha}{2\sin \alpha+\sin 2\alpha}[/tex].

[tex]\tan \frac{\alpha}{2}[/tex]

[tex]\cot ^2\frac{\alpha}{2}[/tex]

[tex]\tan ^2\alpha[/tex]

[tex]\tan ^2\frac{\alpha}{2}[/tex]

Problem 8

The cosines of two angles of a triangle are [tex]\frac{3}{5}[/tex] and [tex]\frac{5}{13}[/tex]. Find the sine of the third angle.

[tex]\frac{33}{65}[/tex]

[tex]\frac{56}{65}[/tex]

[tex]\frac{16}{65}[/tex]

[tex]\frac{63}{65}[/tex]

Problem 9

Find the value of the expression [tex]\cos(x+30^{\circ})\cos(x-30^{\circ})[/tex] if [tex]\cos 2x=-\frac{4}{5}[/tex].

[tex]-\frac{13}{20}[/tex]

[tex]\frac{13}{20}[/tex]

[tex]-\frac{3}{10}[/tex]

[tex]-\frac{3}{20}[/tex]

Problem 10

Calculate the value of the expression [tex]\frac{\cot 32^{\circ}\tan 77^{\circ}-1}{\tan 58^{\circ}+\cot 13^{\circ}}[/tex].

Problem 11

Find the value of the expression [tex]\sin ^4\alpha+\cos^4\alpha[/tex] if [tex]\sin \alpha-\cos\alpha=\frac{1}{2}[/tex].

[tex]\frac{23}{32}[/tex]

[tex]\frac{55}{64}[/tex]

[tex]\frac{1}{4}[/tex]

[tex]\frac{7}{16}[/tex]

Problem 12

If [tex]\sin \alpha=-0. 8[/tex] and [tex]\alpha\in\left(270^{\circ}, 360^{\circ}\right)[/tex], calculate [tex]\sqrt{5}\sin \frac{\alpha}{2}+3\tan \alpha[/tex].

Problem 13

If [tex]\frac{1}{1-\sin x}+\frac{1}{1+\sin x}=6[/tex], find [tex]\tan x[/tex].

[tex]\tan x=\sqrt{2}[/tex]

[tex]\tan x=\pm\sqrt{2}[/tex]

[tex]\tan x=\pm\frac{\sqrt{2}}{2}[/tex]

[tex]\tan x=\pm\sqrt{3}[/tex]

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Trigonometric Expressions and Identities: Problems with Solutions

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