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Trigonometric Identities Practice Questions with Detailed Solutions – Level 2

Intermediate trigonometric identity problems on higher powers, tangent products, shifted maximum and minimum values and eliminating theta, each explained step by step.

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Questions, answer key and solutions.

Questions
50
Duration
60 min
Marks
50

Negative marking

Attempts
100
Access
Free
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About this test

Trigonometric Identities Level 2 Practice Quiz: Powers, Products and Extremes

Level 2 of the trigonometric identities series develops multi-step simplification: combining identities, higher powers of sine and cosine, and extreme values of shifted expressions. Aptitude360 designed this intermediate set for quant aptitude practice, and each question includes a written solution showing the identity used.

What this Level 2 trigonometry set tests

  • Finding cosine or sine from a given sum of secant and tangent, or of cosecant and cotangent
  • Maximum and minimum of sine-cosine combinations with a constant added or subtracted
  • Minimum of expressions that mix secant squared with cosine squared
  • Fourth, third and sixth powers of sine and cosine
  • Long products of tangents and sums of logarithms of tangents
  • Sums of sine squared over angles in arithmetic progression
  • Eliminating theta when two variables are written in secant and tangent

Useful identity rules and formulas for Level 2

  • Square of sum: \( (\sin\theta+\cos\theta)^{2}=1+2\sin\theta\cos\theta \)
  • Fourth powers: \( \sin^{4}\theta+\cos^{4}\theta=1-2\sin^{2}\theta\cos^{2}\theta \)
  • Tan plus cot: \( \tan\theta+\cot\theta=\dfrac{1}{\sin\theta\cos\theta} \)
  • Complementary product: \( \tan\theta\times\tan(90^{\circ}-\theta)=1 \)
  • AM-GM bound for \( a,b>0 \): \( a\sec^{2}\theta+b\cos^{2}\theta\ge2\sqrt{ab} \)

Preparation tips for intermediate identity problems

  • Treat a constant added to a sine-cosine expression as a shift: the range moves up or down by that constant
  • Pair angles from both ends of a long tangent product so each pair multiplies to one
  • Rewrite higher powers using the square of sine squared plus cosine squared
  • When a minus sign sits before a sine-cosine block, the maximum of the whole uses the minimum of the block

Who this set suits

Students who are comfortable with the basic identities and want firmer practice for SSC, banking and other aptitude exams.

FAQs about Trigonometric Identities Practice Questions with Detailed Solutions – Level 2

How do you find sin4 theta plus cos4 theta?
Write it as one minus twice sine squared times cosine squared. This follows from squaring the identity sine squared plus cosine squared equals one.
What is the value of tan 1 times tan 2 up to tan 89?
The product equals one, because tangent of an angle multiplied by tangent of its complement is one and the middle term, tangent forty-five degrees, is also one.
What is the sum of log tan 1 degree to log tan 89 degrees?
The sum is zero, since it equals the logarithm of the product of those tangents, which is one.
How do you find the maximum value of a sin theta plus b cos theta plus c?
Add c to the square root of a squared plus b squared. The minimum is c minus that square root.

Test structure

How the paper is split, and how long each part runs. A section you have finished cannot be returned to.

SectionQuestionsTime
Trigonometric Identities Level 25060 min

Marking & rules

Total marks
50 across 50 questions.
Negative marking
A wrong answer costs marks. Leaving a question unanswered costs nothing.
Attempts
100 in total. An attempt is spent the moment you begin one.

What you get when you submit

The analysis is ready the moment you submit the paper.

Score and accuracy

Marks, percentage, right, wrong and skipped, and the time you took.

Section by section

The same figures for each section, so a weak one cannot hide in an average.

The full answer key

Every question with your answer, the right one, the working, and the video where there is one.

Where you stand

Your score against the spread of everyone else who has sat this test.

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